העתקה ליניארית – ויקיפדיה

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href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_he.wikipedia.org&uselang=he" class=""><span>תרומה לוויקיפדיה</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%94%D7%A8%D7%A9%D7%9E%D7%94_%D7%9C%D7%97%D7%A9%D7%91%D7%95%D7%9F&returnto=%D7%94%D7%A2%D7%AA%D7%A7%D7%94+%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA" title="מומלץ ליצור חשבון ולהיכנס אליו, אך אין חובה לעשות זאת" class=""><span>יצירת חשבון</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%9B%D7%A0%D7%99%D7%A1%D7%94_%D7%9C%D7%97%D7%A9%D7%91%D7%95%D7%9F&returnto=%D7%94%D7%A2%D7%AA%D7%A7%D7%94+%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA" title="מומלץ להיכנס לחשבון, אך אין חובה לעשות זאת [o]" accesskey="o" class=""><span>כניסה לחשבון</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="אפשרויות נוספות" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="כלים אישיים" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">כלים אישיים</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="תפריט משתמש" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_he.wikipedia.org&uselang=he"><span>תרומה לוויקיפדיה</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%94%D7%A8%D7%A9%D7%9E%D7%94_%D7%9C%D7%97%D7%A9%D7%91%D7%95%D7%9F&returnto=%D7%94%D7%A2%D7%AA%D7%A7%D7%94+%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA" title="מומלץ ליצור חשבון ולהיכנס אליו, אך אין חובה לעשות זאת"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>יצירת חשבון</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%9B%D7%A0%D7%99%D7%A1%D7%94_%D7%9C%D7%97%D7%A9%D7%91%D7%95%D7%9F&returnto=%D7%94%D7%A2%D7%AA%D7%A7%D7%94+%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA" title="מומלץ להיכנס לחשבון, אך אין חובה לעשות זאת [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>כניסה לחשבון</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> דפים לעורכים שלא נכנסו לחשבון <a href="/wiki/%D7%A2%D7%96%D7%A8%D7%94:%D7%91%D7%A8%D7%95%D7%9B%D7%99%D7%9D_%D7%94%D7%91%D7%90%D7%99%D7%9D" aria-label="מידע נוסף על עריכה"><span>מידע נוסף</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%94%D7%AA%D7%A8%D7%95%D7%9E%D7%95%D7%AA_%D7%A9%D7%9C%D7%99" title="רשימת העריכות שנעשו מכתובת IP זו [y]" accesskey="y"><span>תרומות</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%94%D7%A9%D7%99%D7%97%D7%94_%D7%A9%D7%9C%D7%99" title="דיון על העריכות שנעשו מכתובת IP זו [n]" accesskey="n"><span>שיחה</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"><div id="mw-dismissablenotice-anonplace"></div><script>(function(){var node=document.getElementById("mw-dismissablenotice-anonplace");if(node){node.outerHTML="\u003Cdiv class=\"mw-dismissable-notice\"\u003E\u003Cdiv class=\"mw-dismissable-notice-close\"\u003E[\u003Ca tabindex=\"0\" role=\"button\"\u003Eהסתרה\u003C/a\u003E]\u003C/div\u003E\u003Cdiv class=\"mw-dismissable-notice-body\"\u003E\u003C!-- CentralNotice --\u003E\u003Cdiv id=\"localNotice\" data-nosnippet=\"\"\u003E\u003Cdiv class=\"anonnotice\" lang=\"he\" dir=\"rtl\"\u003E\u003Cp\u003E\u003Cb\u003Eתמיד רציתם לכתוב בוויקיפדיה אבל לא ידעתם איך? אתם מוזמנים לסדנת עריכה בוויקיפדיה. הסדנה תתקיים בספרייה הלאומית (בבניינה החדש) בירושלים ביום שישי, 06.12.24, בשעה 09:00. להרשמה לחצו \u003Ca href=\"/wiki/%D7%95%D7%99%D7%A7%D7%99%D7%A4%D7%93%D7%99%D7%94:%D7%9E%D7%99%D7%96%D7%9E%D7%99_%D7%95%D7%99%D7%A7%D7%99%D7%A4%D7%93%D7%99%D7%94/%D7%92%D7%9C%D7%90%D7%9D/%D7%94%D7%A1%D7%A4%D7%A8%D7%99%D7%99%D7%94_%D7%94%D7%9C%D7%90%D7%95%D7%9E%D7%99%D7%AA/%D7%90%D7%99%D7%A8%D7%95%D7%A2%D7%99%D7%9D/%D7%A1%D7%93%D7%A0%D7%AA_%D7%A2%D7%A8%D7%99%D7%9B%D7%94_%D7%93%D7%A6%D7%9E%D7%91%D7%A8_2024\" title=\"ויקיפדיה:מיזמי ויקיפדיה/גלאם/הספרייה הלאומית/אירועים/סדנת עריכה דצמבר 2024\"\u003Eכאן\u003C/a\u003E.\u003C/b\u003E\n\u003C/p\u003E\u003C/div\u003E\u003C/div\u003E\u003C/div\u003E\u003C/div\u003E";}}());</script></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="אתר"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="תוכן עניינים" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">תוכן עניינים</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">העברה לסרגל הצד</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">הסתרה</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">התחלה</div> </a> </li> <li id="toc-הגדרה" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#הגדרה"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>הגדרה</span> </div> </a> <ul id="toc-הגדרה-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-דוגמאות_ומקרים_מיוחדים" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#דוגמאות_ומקרים_מיוחדים"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>דוגמאות ומקרים מיוחדים</span> </div> </a> <ul id="toc-דוגמאות_ומקרים_מיוחדים-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-סוגי_העתקות_ליניאריות" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#סוגי_העתקות_ליניאריות"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>סוגי העתקות ליניאריות</span> </div> </a> <ul id="toc-סוגי_העתקות_ליניאריות-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-מרחב_ההעתקות_הליניאריות" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#מרחב_ההעתקות_הליניאריות"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>מרחב ההעתקות הליניאריות</span> </div> </a> <ul id="toc-מרחב_ההעתקות_הליניאריות-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-גרעין_ותמונה_של_העתקה_ליניארית" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#גרעין_ותמונה_של_העתקה_ליניארית"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>גרעין ותמונה של העתקה ליניארית</span> </div> </a> <button aria-controls="toc-גרעין_ותמונה_של_העתקה_ליניארית-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>שינוי מצב התת־פרק גרעין ותמונה של העתקה ליניארית</span> </button> <ul id="toc-גרעין_ותמונה_של_העתקה_ליניארית-sublist" class="vector-toc-list"> <li id="toc-משפט_הדרגה" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#משפט_הדרגה"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>משפט הדרגה</span> </div> </a> <ul id="toc-משפט_הדרגה-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-מטריצה_מייצגת_של_העתקה_ליניארית" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#מטריצה_מייצגת_של_העתקה_ליניארית"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>מטריצה מייצגת של העתקה ליניארית</span> </div> </a> <button aria-controls="toc-מטריצה_מייצגת_של_העתקה_ליניארית-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>שינוי מצב התת־פרק מטריצה מייצגת של העתקה ליניארית</span> </button> <ul id="toc-מטריצה_מייצגת_של_העתקה_ליניארית-sublist" class="vector-toc-list"> <li id="toc-תכונות" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#תכונות"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>תכונות</span> </div> </a> <ul id="toc-תכונות-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-ההעתקה_הדואלית" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#ההעתקה_הדואלית"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>ההעתקה הדואלית</span> </div> </a> <button aria-controls="toc-ההעתקה_הדואלית-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>שינוי מצב התת־פרק ההעתקה הדואלית</span> </button> <ul id="toc-ההעתקה_הדואלית-sublist" class="vector-toc-list"> <li id="toc-תכונות_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#תכונות_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>תכונות</span> </div> </a> <ul id="toc-תכונות_2-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-קישורים_חיצוניים" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#קישורים_חיצוניים"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>קישורים חיצוניים</span> </div> </a> <ul id="toc-קישורים_חיצוניים-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="תוכן עניינים" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="מצב תוכן העניינים" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">מצב תוכן העניינים</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">העתקה ליניארית</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="מעבר לערך בשפה אחרת. זמין ב־49 שפות" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-49" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">49 שפות</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Linear_map" title="Linear map – אנגלית" lang="en" hreflang="en" data-title="Linear map" data-language-autonym="English" data-language-local-name="אנגלית" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D8%AD%D9%88%D9%8A%D9%84_%D8%AE%D8%B7%D9%8A" title="تحويل خطي – ערבית" lang="ar" hreflang="ar" data-title="تحويل خطي" data-language-autonym="العربية" data-language-local-name="ערבית" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9B%D0%B8%D0%BD%D0%B5%D0%B5%D0%BD_%D0%BE%D0%BF%D0%B5%D1%80%D0%B0%D1%82%D0%BE%D1%80" title="Линеен оператор – בולגרית" lang="bg" hreflang="bg" data-title="Линеен оператор" data-language-autonym="Български" data-language-local-name="בולגרית" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Aplicaci%C3%B3_lineal" title="Aplicació lineal – קטלאנית" lang="ca" hreflang="ca" data-title="Aplicació lineal" data-language-autonym="Català" data-language-local-name="קטלאנית" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%86%DB%95%D8%AE%D8%B4%DB%95%DB%8C_%DA%BE%DB%8E%DA%B5%DB%8C" title="نەخشەی ھێڵی – כורדית סוראנית" lang="ckb" hreflang="ckb" data-title="نەخشەی ھێڵی" data-language-autonym="کوردی" data-language-local-name="כורדית סוראנית" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Line%C3%A1rn%C3%AD_zobrazen%C3%AD" title="Lineární zobrazení – צ׳כית" lang="cs" hreflang="cs" data-title="Lineární zobrazení" data-language-autonym="Čeština" data-language-local-name="צ׳כית" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9B%D0%B8%D0%BD%D0%B8%D0%BB%D0%BB%D0%B5_%D0%BA%D1%83%C3%A7%D0%B0%D1%80%D1%83" title="Линилле куçару – צ׳ובשית" lang="cv" hreflang="cv" data-title="Линилле куçару" data-language-autonym="Чӑвашла" data-language-local-name="צ׳ובשית" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Line%C3%A6r_funktion" title="Lineær funktion – דנית" lang="da" hreflang="da" data-title="Lineær funktion" data-language-autonym="Dansk" data-language-local-name="דנית" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Lineare_Abbildung" title="Lineare Abbildung – גרמנית" lang="de" hreflang="de" data-title="Lineare Abbildung" data-language-autonym="Deutsch" data-language-local-name="גרמנית" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%93%CF%81%CE%B1%CE%BC%CE%BC%CE%B9%CE%BA%CE%AE_%CE%B1%CF%80%CE%B5%CE%B9%CE%BA%CF%8C%CE%BD%CE%B9%CF%83%CE%B7" title="Γραμμική απεικόνιση – יוונית" lang="el" hreflang="el" data-title="Γραμμική απεικόνιση" data-language-autonym="Ελληνικά" data-language-local-name="יוונית" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Lineara_transformo" title="Lineara transformo – אספרנטו" lang="eo" hreflang="eo" data-title="Lineara transformo" data-language-autonym="Esperanto" data-language-local-name="אספרנטו" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Aplicaci%C3%B3n_lineal" title="Aplicación lineal – ספרדית" lang="es" hreflang="es" data-title="Aplicación lineal" data-language-autonym="Español" data-language-local-name="ספרדית" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Lineaarkujutus" title="Lineaarkujutus – אסטונית" lang="et" hreflang="et" data-title="Lineaarkujutus" data-language-autonym="Eesti" data-language-local-name="אסטונית" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Aplikazio_lineal" title="Aplikazio lineal – בסקית" lang="eu" hreflang="eu" data-title="Aplikazio lineal" data-language-autonym="Euskara" data-language-local-name="בסקית" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%86%DA%AF%D8%A7%D8%B4%D8%AA_%D8%AE%D8%B7%DB%8C" title="نگاشت خطی – פרסית" lang="fa" hreflang="fa" data-title="نگاشت خطی" data-language-autonym="فارسی" data-language-local-name="פרסית" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Lineaarikuvaus" title="Lineaarikuvaus – פינית" lang="fi" hreflang="fi" data-title="Lineaarikuvaus" data-language-autonym="Suomi" data-language-local-name="פינית" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Application_lin%C3%A9aire" title="Application linéaire – צרפתית" lang="fr" hreflang="fr" data-title="Application linéaire" data-language-autonym="Français" data-language-local-name="צרפתית" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Aplicaci%C3%B3n_linear" title="Aplicación linear – גליסית" lang="gl" hreflang="gl" data-title="Aplicación linear" data-language-autonym="Galego" data-language-local-name="גליסית" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Linearni_operator" title="Linearni operator – קרואטית" lang="hr" hreflang="hr" data-title="Linearni operator" data-language-autonym="Hrvatski" data-language-local-name="קרואטית" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Line%C3%A1ris_lek%C3%A9pez%C3%A9s" title="Lineáris leképezés – הונגרית" lang="hu" hreflang="hu" data-title="Lineáris leképezés" data-language-autonym="Magyar" data-language-local-name="הונגרית" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B3%D5%AE%D5%A1%D5%B5%D5%AB%D5%B6_%D5%A1%D6%80%D5%BF%D5%A1%D5%BA%D5%A1%D5%BF%D5%AF%D5%A5%D6%80%D5%B8%D6%82%D5%B4" title="Գծային արտապատկերում – ארמנית" lang="hy" hreflang="hy" data-title="Գծային արտապատկերում" data-language-autonym="Հայերեն" data-language-local-name="ארמנית" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Peta_linear" title="Peta linear – אינדונזית" lang="id" hreflang="id" data-title="Peta linear" data-language-autonym="Bahasa Indonesia" data-language-local-name="אינדונזית" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/L%C3%ADnuleg_v%C3%B6rpun" title="Línuleg vörpun – איסלנדית" lang="is" hreflang="is" data-title="Línuleg vörpun" data-language-autonym="Íslenska" data-language-local-name="איסלנדית" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Trasformazione_lineare" title="Trasformazione lineare – איטלקית" lang="it" hreflang="it" data-title="Trasformazione lineare" data-language-autonym="Italiano" data-language-local-name="איטלקית" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%B7%9A%E5%9E%8B%E5%86%99%E5%83%8F" title="線型写像 – יפנית" lang="ja" hreflang="ja" data-title="線型写像" data-language-autonym="日本語" data-language-local-name="יפנית" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%84%A0%ED%98%95_%EB%B3%80%ED%99%98" title="선형 변환 – קוריאנית" lang="ko" hreflang="ko" data-title="선형 변환" data-language-autonym="한국어" data-language-local-name="קוריאנית" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Tiesinis_operatorius" title="Tiesinis operatorius – ליטאית" lang="lt" hreflang="lt" data-title="Tiesinis operatorius" data-language-autonym="Lietuvių" data-language-local-name="ליטאית" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Lineaire_afbeelding" title="Lineaire afbeelding – הולנדית" lang="nl" hreflang="nl" data-title="Lineaire afbeelding" data-language-autonym="Nederlands" data-language-local-name="הולנדית" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Line%C3%A6r_transformasjon" title="Lineær transformasjon – נורווגית ספרותית" lang="nb" hreflang="nb" data-title="Lineær transformasjon" data-language-autonym="Norsk bokmål" data-language-local-name="נורווגית ספרותית" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B2%E0%A9%80%E0%A8%A8%E0%A9%80%E0%A8%85%E0%A8%B0_%E0%A8%AE%E0%A9%88%E0%A8%AA" title="ਲੀਨੀਅਰ ਮੈਪ – פנג׳אבי" lang="pa" hreflang="pa" data-title="ਲੀਨੀਅਰ ਮੈਪ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="פנג׳אבי" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Przekszta%C5%82cenie_liniowe" title="Przekształcenie liniowe – פולנית" lang="pl" hreflang="pl" data-title="Przekształcenie liniowe" data-language-autonym="Polski" data-language-local-name="פולנית" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D9%84%DB%8C%D9%86%DB%8C%D8%A6%D8%B1_%D9%85%DB%8C%D9%BE" title="لینیئر میپ – פנג׳בית מערבית" lang="pnb" hreflang="pnb" data-title="لینیئر میپ" data-language-autonym="پنجابی" data-language-local-name="פנג׳בית מערבית" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Transforma%C3%A7%C3%A3o_linear" title="Transformação linear – פורטוגזית" lang="pt" hreflang="pt" data-title="Transformação linear" data-language-autonym="Português" data-language-local-name="פורטוגזית" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Transformare_liniar%C4%83" title="Transformare liniară – רומנית" lang="ro" hreflang="ro" data-title="Transformare liniară" data-language-autonym="Română" data-language-local-name="רומנית" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9B%D0%B8%D0%BD%D0%B5%D0%B9%D0%BD%D0%BE%D0%B5_%D0%BE%D1%82%D0%BE%D0%B1%D1%80%D0%B0%D0%B6%D0%B5%D0%BD%D0%B8%D0%B5" title="Линейное отображение – רוסית" lang="ru" hreflang="ru" data-title="Линейное отображение" data-language-autonym="Русский" data-language-local-name="רוסית" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Linearna_transformacija" title="Linearna transformacija – סרבו-קרואטית" lang="sh" hreflang="sh" data-title="Linearna transformacija" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="סרבו-קרואטית" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Linear_mapping" title="Linear mapping – אנגלית פשוטה" lang="en-simple" hreflang="en-simple" data-title="Linear mapping" data-language-autonym="Simple English" data-language-local-name="אנגלית פשוטה" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Line%C3%A1rne_zobrazenie" title="Lineárne zobrazenie – סלובקית" lang="sk" hreflang="sk" data-title="Lineárne zobrazenie" data-language-autonym="Slovenčina" data-language-local-name="סלובקית" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Linearna_transformacija" title="Linearna transformacija – סלובנית" lang="sl" hreflang="sl" data-title="Linearna transformacija" data-language-autonym="Slovenščina" data-language-local-name="סלובנית" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Harta_lineare" title="Harta lineare – אלבנית" lang="sq" hreflang="sq" data-title="Harta lineare" data-language-autonym="Shqip" data-language-local-name="אלבנית" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9B%D0%B8%D0%BD%D0%B5%D0%B0%D1%80%D0%BD%D0%BE_%D0%BF%D1%80%D0%B5%D1%81%D0%BB%D0%B8%D0%BA%D0%B0%D0%B2%D0%B0%D1%9A%D0%B5" title="Линеарно пресликавање – סרבית" lang="sr" hreflang="sr" data-title="Линеарно пресликавање" data-language-autonym="Српски / srpski" data-language-local-name="סרבית" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Linj%C3%A4r_avbildning" title="Linjär avbildning – שוודית" lang="sv" hreflang="sv" data-title="Linjär avbildning" data-language-autonym="Svenska" data-language-local-name="שוודית" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%A8%E0%AF%87%E0%AE%B0%E0%AE%BF%E0%AE%AF%E0%AE%B2%E0%AF%8D_%E0%AE%95%E0%AF%8B%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AF%81" title="நேரியல் கோப்பு – טמילית" lang="ta" hreflang="ta" data-title="நேரியல் கோப்பு" data-language-autonym="தமிழ்" data-language-local-name="טמילית" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Do%C4%9Frusal_d%C3%B6n%C3%BC%C5%9F%C3%BCm" title="Doğrusal dönüşüm – טורקית" lang="tr" hreflang="tr" data-title="Doğrusal dönüşüm" data-language-autonym="Türkçe" data-language-local-name="טורקית" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9B%D1%96%D0%BD%D1%96%D0%B9%D0%BD%D0%B5_%D0%B2%D1%96%D0%B4%D0%BE%D0%B1%D1%80%D0%B0%D0%B6%D0%B5%D0%BD%D0%BD%D1%8F" title="Лінійне відображення – אוקראינית" lang="uk" hreflang="uk" data-title="Лінійне відображення" data-language-autonym="Українська" data-language-local-name="אוקראינית" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%84%DA%A9%DB%8C%D8%B1%DB%8C_%D8%A7%D8%B3%D8%AA%D8%AD%D8%A7%D9%84%DB%81" title="لکیری استحالہ – אורדו" lang="ur" hreflang="ur" data-title="لکیری استحالہ" data-language-autonym="اردو" data-language-local-name="אורדו" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Bi%E1%BA%BFn_%C4%91%E1%BB%95i_tuy%E1%BA%BFn_t%C3%ADnh" title="Biến đổi tuyến tính – וייטנאמית" lang="vi" hreflang="vi" data-title="Biến đổi tuyến tính" data-language-autonym="Tiếng Việt" data-language-local-name="וייטנאמית" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84" title="线性映射 – סינית וו" lang="wuu" hreflang="wuu" data-title="线性映射" data-language-autonym="吴语" data-language-local-name="סינית וו" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84" title="线性映射 – סינית" lang="zh" hreflang="zh" data-title="线性映射" data-language-autonym="中文" data-language-local-name="סינית" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q207643#sitelinks-wikipedia" title="עריכת קישורים בין־לשוניים" class="wbc-editpage">עריכת הקישורים</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="מרחבי שם"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/%D7%94%D7%A2%D7%AA%D7%A7%D7%94_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA" title="צפייה בדף התוכן [c]" 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href="/w/index.php?title=%D7%9E%D7%99%D7%95%D7%97%D7%93:QrCode&url=https%3A%2F%2Fhe.wikipedia.org%2Fwiki%2F%25D7%2594%25D7%25A2%25D7%25AA%25D7%25A7%25D7%2594_%25D7%259C%25D7%2599%25D7%25A0%25D7%2599%25D7%2590%25D7%25A8%25D7%2599%25D7%25AA"><span>הורדת קוד QR</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> הדפסה/יצוא </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%A1%D7%A4%D7%A8&bookcmd=book_creator&referer=%D7%94%D7%A2%D7%AA%D7%A7%D7%94+%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA"><span>יצירת ספר</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=%D7%9E%D7%99%D7%95%D7%97%D7%93:DownloadAsPdf&page=%D7%94%D7%A2%D7%AA%D7%A7%D7%94_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA&action=show-download-screen"><span>הורדה כ־PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=%D7%94%D7%A2%D7%AA%D7%A7%D7%94_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA&printable=yes" title="גרסה להדפסה של הדף הזה [p]" accesskey="p"><span>גרסה להדפסה</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> במיזמים אחרים </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Linear_operators" hreflang="en"><span>ויקישיתוף</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a 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class="vector-pinnable-header-label">מראה</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">העברה לסרגל הצד</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">הסתרה</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">מתוך ויקיפדיה, האנציקלופדיה החופשית</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">(הופנה מהדף <a href="/w/index.php?title=%D7%98%D7%A8%D7%A0%D7%A1%D7%A4%D7%95%D7%A8%D7%9E%D7%A6%D7%99%D7%94_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA&redirect=no" class="mw-redirect" title="טרנספורמציה ליניארית">טרנספורמציה ליניארית</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-rtl mw-parser-output" lang="he" dir="rtl"><table cellpadding="1" style="float: left; clear:both; border: 1px solid #8888aa; background: #f7f8ff; padding: 5px; font-size: 95%; margin: 0.5em 1em 0.5em 0.5em;"> <tbody><tr> <td style="text-align: center;"><span typeof="mw:File"><a href="/wiki/%D7%A1%D7%99%D7%9E%D7%95%D7%9F_%D7%9E%D7%AA%D7%9E%D7%98%D7%99" title="סימון מתמטי"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/60px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/90px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/120px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span> <p>בערך זה<br />נעשה שימוש<br />בסימנים מוסכמים<br />מתחום המתמטיקה.<br />להבהרת הסימנים<br />ראו <a href="/wiki/%D7%A1%D7%99%D7%9E%D7%95%D7%9F_%D7%9E%D7%AA%D7%9E%D7%98%D7%99" title="סימון מתמטי">סימון מתמטי</a>. </p> </td></tr></tbody></table> <p>ב<a href="/wiki/%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA" title="אלגברה ליניארית">אלגברה ליניארית</a>, <b>העתקה ליניארית</b> או <b>טרנספורמציה ליניארית</b>, היא העתקה (פונקציה) מ<a href="/wiki/%D7%9E%D7%A8%D7%97%D7%91_%D7%95%D7%A7%D7%98%D7%95%D7%A8%D7%99" title="מרחב וקטורי">מרחב וקטורי</a> למרחב וקטורי, השומרת על החיבור וה<a href="/wiki/%D7%9B%D7%A4%D7%9C_%D7%91%D7%A1%D7%A7%D7%9C%D7%A8" title="כפל בסקלר">כפל בסקלר</a>. באופן פורמלי יותר, העתקה ליניארית היא <a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94" title="פונקציה">העתקה</a> <a href="/wiki/%D7%90%D7%93%D7%99%D7%98%D7%99%D7%91%D7%99%D7%95%D7%AA" class="mw-redirect" title="אדיטיביות">אדיטיבית</a> ו<a href="/wiki/%D7%94%D7%95%D7%9E%D7%95%D7%92%D7%A0%D7%99%D7%95%D7%AA_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" class="mw-redirect" title="הומוגניות (מתמטיקה)">הומוגנית</a> בין שני מרחבים וקטוריים (מעל אותו <a href="/wiki/%D7%A9%D7%93%D7%94_(%D7%9E%D7%91%D7%A0%D7%94_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99)" title="שדה (מבנה אלגברי)">שדה</a>). מכיוון שהעתקה ליניארית שומרת על כל הפעולות, היא מהווה <a href="/wiki/%D7%94%D7%95%D7%9E%D7%95%D7%9E%D7%95%D7%A8%D7%A4%D7%99%D7%96%D7%9D_(%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94)" class="mw-redirect" title="הומומורפיזם (אלגברה)">מורפיזם</a> בקטגוריה של המרחבים מעל השדה. </p><p>העתקה בין מרחבים מממד סופי אפשר לתאר באמצעות <a href="/wiki/%D7%9E%D7%98%D7%A8%D7%99%D7%A6%D7%94" title="מטריצה">מטריצה</a>; כל מטריצה מתארת באופן חד-משמעי העתקה ליניארית, וכל העתקה ליניארית ניתנת לייצוג ככפל של מטריצה בוקטורי קואורדינטות (באופן פורמלי: מרחב ההעתקות ומרחב המטריצות <a href="/wiki/%D7%90%D7%99%D7%96%D7%95%D7%9E%D7%95%D7%A8%D7%A4%D7%99%D7%96%D7%9D" title="איזומורפיזם">איזומורפיים</a>). </p><p>להעתקה ליניארית ממרחב <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> אל עצמו, כלומר <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\colon V\to V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>:</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T\colon V\to V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3a6ee7fc18a25eec79f8b0e2adff07caea23a9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.859ex; height:2.176ex;" alt="{\displaystyle T\colon V\to V}"></span>, נהוג לעיתים לקרוא <b><a href="/wiki/%D7%90%D7%95%D7%A4%D7%A8%D7%98%D7%95%D7%A8" title="אופרטור">אופרטור</a> ליניארי</b>, אך המושג אופרטור ליניארי משמש גם לתיאור העתקה ליניארית כלשהי. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="הגדרה"><span id=".D7.94.D7.92.D7.93.D7.A8.D7.94"></span>הגדרה</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%94%D7%A2%D7%AA%D7%A7%D7%94_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA&action=edit&section=1" title="עריכת קוד המקור של הפרק: הגדרה"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%94%D7%A2%D7%AA%D7%A7%D7%94_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA&veaction=edit&section=1" title="עריכת פסקה: "הגדרה"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>העתקה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> ממרחב וקטורי <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> מעל שדה F אל מרחב וקטורי <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span> מעל אותו שדה (מסמנים <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\colon V\to W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>:</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T\colon V\to W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9307c45fa2557b519d013c5b4229c9bd53a6c3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.507ex; height:2.176ex;" alt="{\displaystyle T\colon V\to W}"></span>) תקרא <b>העתקה ליניארית</b> או <b>טרנספורמציה ליניארית</b>, אם מתקיימים התנאים הבאים: </p> <dl><dd><ol><li><b><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> משמרת חיבור (<a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94_%D7%90%D7%93%D7%99%D7%98%D7%99%D7%91%D7%99%D7%AA" title="פונקציה אדיטיבית">אדיטיביות</a>)</b>: לכל שני <a href="/wiki/%D7%95%D7%A7%D7%98%D7%95%D7%A8_(%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94)" class="mw-redirect" title="וקטור (אלגברה)">וקטורים</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v,u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>,</mo> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v,u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70e0ca9d5128993f768b92ebee01a8286e65d7e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.491ex; height:2.009ex;" alt="{\displaystyle v,u}"></span> השייכים למרחב <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> מתקיים: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(v+u)=T(v)+T(u)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo>+</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(v+u)=T(v)+T(u)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb058923428acb2f2bb978e4f7ea89954a4b0d43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.031ex; height:2.843ex;" alt="{\displaystyle T(v+u)=T(v)+T(u)}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> <b>משמרת כפל בסקלר (<a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94_%D7%94%D7%95%D7%9E%D7%95%D7%92%D7%A0%D7%99%D7%AA" title="פונקציה הומוגנית">הומוגניות</a>)</b>: לכל וקטור <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> השייך למרחב <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span>, ולכל <a href="/wiki/%D7%A1%D7%A7%D7%9C%D7%A8_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="סקלר (מתמטיקה)">סקלר</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> השייך לשדה F מתקיים: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(\alpha v)=\alpha T(v)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>α</mi> <mi>T</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(\alpha v)=\alpha T(v)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2392f53569e007bce8287aea44f6d6d40ebc4852" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.22ex; height:2.843ex;" alt="{\displaystyle T(\alpha v)=\alpha T(v)}"></span></li></ol></dd></dl> <p>משמעות התנאים הללו היא שאין זה משנה אם מפעילים את ההעתקה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> (אשר מניבה את תמונת הפונקציה) על כל וקטור בנפרד ואחר כך מחברים את התמונות, או שמחברים את הווקטורים <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v,u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>,</mo> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v,u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70e0ca9d5128993f768b92ebee01a8286e65d7e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.491ex; height:2.009ex;" alt="{\displaystyle v,u}"></span> ולאחר מכן מפעילים על הסכום את העתקה – התוצאה תהיה זהה, דהיינו נשמר החיבור (אדיטיביות). באותו אופן, אין זה משנה אם מפעילים את ההעתקה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> על התוצאה של כפל הווקטור <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> בסקלר <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span>, או שמפעילים את ההעתקה על הווקטור <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> ולאחר מכן כופלים את התמונה בסקלר <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> – הכפל נשמר (הומוגניות). שתי תכונות אלו מרכיבות את תכונת הליניאריות. </p><p>מההגדרה נובעת התכונה הכללית, שבהינתן וקטורים <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{v_{k}\}_{k=1}^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msubsup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{v_{k}\}_{k=1}^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c67da6706972f8cf8a45d5eb1f70934bdfea1d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.731ex; height:3.009ex;" alt="{\displaystyle \{v_{k}\}_{k=1}^{n}}"></span> וסקלרים <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\lambda _{k}\}_{k=1}^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msubsup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\lambda _{k}\}_{k=1}^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c224619fabe98c5ceb1101c8d01fb72666b3a2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.958ex; height:3.009ex;" alt="{\displaystyle \{\lambda _{k}\}_{k=1}^{n}}"></span> מתקיים: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ T\left(\sum _{k=1}^{n}{\lambda _{k}v_{k}}\right)=\sum _{k=1}^{n}{\lambda _{k}{T(v_{k})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>T</mi> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ T\left(\sum _{k=1}^{n}{\lambda _{k}v_{k}}\right)=\sum _{k=1}^{n}{\lambda _{k}{T(v_{k})}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/229f6c27d10816b7b2001f55d03b7995d1cea7fb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:29.634ex; height:7.509ex;" alt="{\displaystyle \ T\left(\sum _{k=1}^{n}{\lambda _{k}v_{k}}\right)=\sum _{k=1}^{n}{\lambda _{k}{T(v_{k})}}}"></span> </p><p>מסקנה נוספת אשר נובעת מההגדרה, היא שה<a href="/wiki/%D7%92%D7%A8%D7%A2%D7%99%D7%9F_(%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94)" title="גרעין (אלגברה)">גרעין</a> של העתקה ליניארית לעולם אינו <a href="/wiki/%D7%94%D7%A7%D7%91%D7%95%D7%A6%D7%94_%D7%94%D7%A8%D7%99%D7%A7%D7%94" title="הקבוצה הריקה">הקבוצה הריקה</a> ותמיד כולל את וקטור האפס <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {0}}\in V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>0</mn> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>∈</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {0}}\in V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98ebd28fb9ac5bfafacd557dde822e410f1939c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.79ex; height:2.843ex;" alt="{\displaystyle {\vec {0}}\in V}"></span>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ T({\vec {0}})=T(0\cdot v)=0\cdot T(v)={\vec {0}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>T</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>0</mn> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>⋅</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>⋅</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>0</mn> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ T({\vec {0}})=T(0\cdot v)=0\cdot T(v)={\vec {0}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd1703350855e2c3f1ccd1ffb5fa93f3436e4c97" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.476ex; height:3.343ex;" alt="{\displaystyle \ T({\vec {0}})=T(0\cdot v)=0\cdot T(v)={\vec {0}}}"></span> </p><p>מסקנה נוספת מכך היא קריטריון הכרחי לכך שפונקציה בין מרחבים וקטוריים היא העתקה ליניארית. אם <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\colon V\to W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>:</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T\colon V\to W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9307c45fa2557b519d013c5b4229c9bd53a6c3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.507ex; height:2.176ex;" alt="{\displaystyle T\colon V\to W}"></span> היא העתקה ליניארית אז <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T({\vec {0}})={\vec {0}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>0</mn> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>0</mn> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T({\vec {0}})={\vec {0}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aea6c6431f67029e00b5b5f84cdff80329b4c4f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.869ex; height:3.343ex;" alt="{\displaystyle T({\vec {0}})={\vec {0}}}"></span>. מאחר שקל לבדוק תנאי זה הוא מהווה כלי יעיל כדי לשלול את היותה של פונקציה חשודה העתקה ליניארית: אם <i>T</i> לא מעבירה אפס לאפס אז היא לא העתקה ליניארית. </p> <div class="mw-heading mw-heading2"><h2 id="דוגמאות_ומקרים_מיוחדים"><span id=".D7.93.D7.95.D7.92.D7.9E.D7.90.D7.95.D7.AA_.D7.95.D7.9E.D7.A7.D7.A8.D7.99.D7.9D_.D7.9E.D7.99.D7.95.D7.97.D7.93.D7.99.D7.9D"></span>דוגמאות ומקרים מיוחדים</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%94%D7%A2%D7%AA%D7%A7%D7%94_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA&action=edit&section=2" title="עריכת קוד המקור של הפרק: דוגמאות ומקרים מיוחדים"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%94%D7%A2%D7%AA%D7%A7%D7%94_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA&veaction=edit&section=2" title="עריכת פסקה: "דוגמאות ומקרים מיוחדים"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>אם <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> היא <a href="/wiki/%D7%9E%D7%98%D7%A8%D7%99%D7%A6%D7%94" title="מטריצה">מטריצה</a> מסדר <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12b23d207d23dd430b93320539abbb0bde84870d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:1.676ex;" alt="{\displaystyle m\times n}"></span>, אז <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> מגדירה העתקה ליניארית <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{A}\colon \mathbb {F} ^{n}\to \mathbb {F} ^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{A}\colon \mathbb {F} ^{n}\to \mathbb {F} ^{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45b91f00a440bcf39a0b35970522700545d06ae6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.204ex; height:2.676ex;" alt="{\displaystyle T_{A}\colon \mathbb {F} ^{n}\to \mathbb {F} ^{m}}"></span> מ-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {F} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6abc33ef38d7d1214109b904b767b7621c100d2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.639ex; height:2.343ex;" alt="{\displaystyle \mathbb {F} ^{n}}"></span> ל-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} ^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {F} ^{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c6f63cb4e1923d7dc2462558ebfc1d10dc22e80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.095ex; height:2.343ex;" alt="{\displaystyle \mathbb {F} ^{m}}"></span> כאשר היא פועלת על <a href="/wiki/%D7%95%D7%A7%D7%98%D7%95%D7%A8_%D7%A2%D7%9E%D7%95%D7%93%D7%94" title="וקטור עמודה">וקטורי עמודה</a> ב־<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {F} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6abc33ef38d7d1214109b904b767b7621c100d2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.639ex; height:2.343ex;" alt="{\displaystyle \mathbb {F} ^{n}}"></span> על ידי <a href="/wiki/%D7%9B%D7%A4%D7%9C_%D7%9E%D7%98%D7%A8%D7%99%D7%A6%D7%95%D7%AA" title="כפל מטריצות">כפל מטריצות</a> מימין: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{A}({\vec {x}})=A{\vec {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{A}({\vec {x}})=A{\vec {x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e53f184dbe5d750e27429c2ed1d7ff31dd1bf420" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.133ex; height:2.843ex;" alt="{\displaystyle T_{A}({\vec {x}})=A{\vec {x}}}"></span>. זוהי דוגמה חשובה ושימושית ביותר, מכיוון שבדרך זו ניתן לייצג כל העתקה ליניארית בין מרחבים מ<a href="/wiki/%D7%9E%D7%9E%D7%93_(%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA)" title="ממד (אלגברה ליניארית)">ממד</a> סופי.</li> <li>טרנספורמציית האפס <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {0} \colon V\to W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo>:</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {0} \colon V\to W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d52e4b64c97aa148c22b0fd380ff7c0783088ac8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.207ex; height:2.176ex;" alt="{\displaystyle \mathbf {0} \colon V\to W}"></span> (פונקציה המתאימה לכל איבר בתחום את <a href="/wiki/%D7%90%D7%99%D7%91%D7%A8_%D7%94%D7%90%D7%A4%D7%A1" title="איבר האפס">וקטור האפס</a> בטווח) ו<a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%99%D7%AA_%D7%94%D7%96%D7%94%D7%95%D7%AA" title="פונקציית הזהות">טרנספורמציית הזהות</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Id} \colon V\to V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Id</mi> <mo>:</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Id} \colon V\to V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88af36f9601742b57d7e185f4443d1daa6903c50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.355ex; height:2.176ex;" alt="{\displaystyle \operatorname {Id} \colon V\to V}"></span> (פונקציה המתאימה לכל איבר בתחום את עצמו) הן טרנספורמציות ליניאריות. בפרט, אם <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=\mathbb {R} ^{n},W=\mathbb {R} ^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>,</mo> <mi>W</mi> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=\mathbb {R} ^{n},W=\mathbb {R} ^{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f603e27ad1861521ef5668ab23b9a9e5688b9d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.703ex; height:2.676ex;" alt="{\displaystyle V=\mathbb {R} ^{n},W=\mathbb {R} ^{m}}"></span> אז את טרנספורמציית האפס ניתן לייצג כ-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/416a1dd7d799d8a81370d796e350420774dc2369" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.822ex; height:2.509ex;" alt="{\displaystyle T_{A}}"></span> (העתקה ליניארית באמצעות הכפלה במטריצה) כאשר <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> היא <a href="/wiki/%D7%9E%D7%98%D7%A8%D7%99%D7%A6%D7%AA_%D7%90%D7%A4%D7%A1%D7%99%D7%9D" title="מטריצת אפסים">מטריצת האפס</a> (מטריצה בגודל המתאים שכולה אפסים), ואת טרנספורמציית הזהות <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Id} \colon \mathbb {R} ^{n}\to \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Id</mi> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Id} \colon \mathbb {R} ^{n}\to \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbdb542cb097212f9174fb84e85defb4ae4bf556" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.573ex; height:2.343ex;" alt="{\displaystyle \operatorname {Id} \colon \mathbb {R} ^{n}\to \mathbb {R} ^{n}}"></span> ניתן לייצג כ-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/416a1dd7d799d8a81370d796e350420774dc2369" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.822ex; height:2.509ex;" alt="{\displaystyle T_{A}}"></span> על ידי <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=I_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=I_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a4babde55611bdac6e8885a70b6774c65eee856" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.083ex; height:2.509ex;" alt="{\displaystyle A=I_{n}}"></span> כאשר <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aba34f081d776e30204f3458e4f50b403b09e5c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.242ex; height:2.509ex;" alt="{\displaystyle I_{n}}"></span> היא <a href="/wiki/%D7%9E%D7%98%D7%A8%D7%99%D7%A6%D7%AA_%D7%94%D7%99%D7%97%D7%99%D7%93%D7%94" title="מטריצת היחידה">מטריצת היחידה</a> מסדר <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> (כלומר, בגודל <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}"></span>).</li> <li>ההעתקה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{A}\colon \mathbb {R} ^{2}\to \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{A}\colon \mathbb {R} ^{2}\to \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f96ca4993ae3ba3d4f76f21cc3da79564f8e7231" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.935ex; height:3.009ex;" alt="{\displaystyle T_{A}\colon \mathbb {R} ^{2}\to \mathbb {R} ^{2}}"></span> עם <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\begin{bmatrix}2&0\\0&1\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\begin{bmatrix}2&0\\0&1\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0af2998c2c35192554bfad9cc568a905499fb0cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:12.696ex; height:6.176ex;" alt="{\displaystyle A={\begin{bmatrix}2&0\\0&1\end{bmatrix}}}"></span> היא העתקה ליניארית המותחת את ציר ה-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> פי <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}"></span>, בעוד את ציר ה-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> היא משאירה ללא שינוי. נבטא אותה במפורש: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}u\\v\end{bmatrix}}={\begin{bmatrix}2&0\\0&1\end{bmatrix}}{\begin{bmatrix}x\\y\end{bmatrix}}={\begin{bmatrix}2x\\y\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>u</mi> </mtd> </mtr> <mtr> <mtd> <mi>v</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>2</mn> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}u\\v\end{bmatrix}}={\begin{bmatrix}2&0\\0&1\end{bmatrix}}{\begin{bmatrix}x\\y\end{bmatrix}}={\begin{bmatrix}2x\\y\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a744ed7dfee07b53dfaf7014cd9f68ae18a09a9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.822ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}u\\v\end{bmatrix}}={\begin{bmatrix}2&0\\0&1\end{bmatrix}}{\begin{bmatrix}x\\y\end{bmatrix}}={\begin{bmatrix}2x\\y\end{bmatrix}}}"></span> ולכן <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{A}(x,y)=(2x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{A}(x,y)=(2x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9753f8487ebf51c8be6512ba85c2ff2bed256bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.74ex; height:2.843ex;" alt="{\displaystyle T_{A}(x,y)=(2x,y)}"></span>. קל לבדוק ישירות שהיא אכן ליניארית. ראו המחשה גרפית שלה באיורים שבתחתית סעיף זה.</li> <li><a href="/wiki/%D7%9E%D7%98%D7%A8%D7%99%D7%A6%D7%AA_%D7%A1%D7%99%D7%91%D7%95%D7%91" title="מטריצת סיבוב">טרנספורמציות סיבוב</a> ו<a href="/wiki/%D7%A9%D7%99%D7%A7%D7%95%D7%A3_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="שיקוף (מתמטיקה)">שיקוף</a> הן טרנספורמציות ליניאריות. לדוגמה, ב-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}"></span>, הטרנספורמציה המשקפת כל וקטור יחסית לציר ה-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> היא טרנספורמציה ליניארית.</li> <li><a href="/wiki/%D7%A0%D7%92%D7%96%D7%A8%D7%AA" title="נגזרת">גזירה</a> היא העתקה ליניארית ממרחב הפונקציות הגזירות למרחב הפונקציות (מרחבים מ<a href="/wiki/%D7%9E%D7%9E%D7%93_(%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA)" title="ממד (אלגברה ליניארית)">ממד</a> אינסופי).</li> <li>פונקציה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon \mathbb {R} \to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon \mathbb {R} \to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1cacd5f7bbe1027cc75fbe2fbd9cb5e79485302" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.283ex; height:2.509ex;" alt="{\displaystyle f\colon \mathbb {R} \to \mathbb {R} }"></span> היא העתקה ליניארית <a href="/wiki/%D7%90%D7%9D_%D7%95%D7%A8%D7%A7_%D7%90%D7%9D" title="אם ורק אם">אם ורק אם</a> היא מהצורה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\alpha x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>α</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=\alpha x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa65f0697bf36f4c4b6f60b34e66c2c8d0988ceb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.333ex; height:2.843ex;" alt="{\displaystyle f(x)=\alpha x}"></span> באשר <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7988141e89a37e7f4deb883dbd74d9bbd6d11317" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.006ex; height:2.176ex;" alt="{\displaystyle \alpha \in \mathbb {R} }"></span>. נשים לב שפונקציה ליניארית (כלומר: כזאת המתארת <a href="/wiki/%D7%A7%D7%95_%D7%99%D7%A9%D7%A8" class="mw-redirect" title="קו ישר">קו ישר</a> ב<a href="/wiki/%D7%9E%D7%99%D7%A9%D7%95%D7%A8_(%D7%92%D7%90%D7%95%D7%9E%D7%98%D7%A8%D7%99%D7%94)" title="מישור (גאומטריה)">מישור האוקלידי</a>) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)=\alpha x+\beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>α</mi> <mi>x</mi> <mo>+</mo> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)=\alpha x+\beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e38abe559a937571e830c6b7565ccfd16de20bc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.343ex; height:2.843ex;" alt="{\displaystyle g(x)=\alpha x+\beta }"></span> היא העתקה ליניארית אם ורק אם <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60b5e78663eba7ba08e0dd4915251e6261f4f35c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.593ex; height:2.509ex;" alt="{\displaystyle \beta =0}"></span>. קל לראות שאחרת <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span> לא מעבירה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {0}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>0</mn> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {0}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e76498919cf387316fc79d04120c59a8d430ef36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.843ex;" alt="{\displaystyle {\vec {0}}}"></span> ל-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {0}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>0</mn> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {0}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e76498919cf387316fc79d04120c59a8d430ef36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.843ex;" alt="{\displaystyle {\vec {0}}}"></span> (תנאי הכרחי להעתקה ליניארית) אך ניתן להוכיח זאת גם באופן יותר מפורש: נניח ש-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta \neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta \neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6347ae2e04e6c0cb5c10d5ec3266a67f9e54da6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.593ex; height:2.676ex;" alt="{\displaystyle \beta \neq 0}"></span> ונראה ש-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span> לא מקיימת אדיטיביות (ליניאריות): <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\alpha +\beta =g(2)=g(1+1)\neq g(1)+g(1)=(\alpha +\beta )+(\alpha +\beta )=2\alpha +2\beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>≠</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mi>α</mi> <mo>+</mo> <mn>2</mn> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\alpha +\beta =g(2)=g(1+1)\neq g(1)+g(1)=(\alpha +\beta )+(\alpha +\beta )=2\alpha +2\beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0db2e827ac249e61c39e0f2bdf7d4b72b64ce23a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:71.273ex; height:2.843ex;" alt="{\displaystyle 2\alpha +\beta =g(2)=g(1+1)\neq g(1)+g(1)=(\alpha +\beta )+(\alpha +\beta )=2\alpha +2\beta }"></span> פונקציה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span> כזאת נקראת <a href="/wiki/%D7%94%D7%A2%D7%AA%D7%A7%D7%94_%D7%90%D7%A4%D7%99%D7%A0%D7%99%D7%AA" title="העתקה אפינית">העתקה אפינית</a>. נעיר שלהעתקות אפיניות יש חשיבות מיוחדת כשדנים בגזירת פונקציות בכמה משתנים (פונקציות <a href="/wiki/%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%91%D7%99%D7%9C%D7%99%D7%95%D7%AA" class="mw-redirect" title="דיפרנציאביליות">דיפרנציאביליות</a>).</li></ul> <ul class="gallery mw-gallery-traditional"> <li class="gallerybox" style="width: 335px"> <div class="thumb" style="width: 330px; height: 230px;"><span typeof="mw:File"><a href="/wiki/%D7%A7%D7%95%D7%91%D7%A5:Streckung_eines_Vektors.gif" class="mw-file-description" title="הפונקציה '"`UNIQ--postMath-00000046-QINU`"' שמוגדרת על-ידי '"`UNIQ--postMath-00000047-QINU`"' היא העתקה ליניארית. פונקציה זו מותחת את רכיב ה-'"`UNIQ--postMath-00000048-QINU`"' בפקטור '"`UNIQ--postMath-00000049-QINU`"'."><img alt="הפונקציה '"`UNIQ--postMath-00000046-QINU`"' שמוגדרת על-ידי '"`UNIQ--postMath-00000047-QINU`"' היא העתקה ליניארית. פונקציה זו מותחת את רכיב ה-'"`UNIQ--postMath-00000048-QINU`"' בפקטור '"`UNIQ--postMath-00000049-QINU`"'." src="//upload.wikimedia.org/wikipedia/commons/thumb/4/43/Streckung_eines_Vektors.gif/259px-Streckung_eines_Vektors.gif" decoding="async" width="259" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/4/43/Streckung_eines_Vektors.gif 1.5x" data-file-width="330" data-file-height="255" /></a></span></div> <div class="gallerytext">הפונקציה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f:\mathbb {R} ^{2}\to \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle f:\mathbb {R} ^{2}\to \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c06d7fc6902e01c04ce84cf57e41c7bd122f1556" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.294ex; height:3.009ex;" alt="{\textstyle f:\mathbb {R} ^{2}\to \mathbb {R} ^{2}}"></span> שמוגדרת על-ידי <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f(x,y)=(2x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle f(x,y)=(2x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dba4d012acc208d82dc8599aa34fa1823f4e6b33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.196ex; height:2.843ex;" alt="{\textstyle f(x,y)=(2x,y)}"></span> היא העתקה ליניארית. פונקציה זו מותחת את רכיב ה-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d951e0f3b54b6a3d73bb9a0a005749046cbce781" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\textstyle x}"></span> בפקטור <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78ed0cd8140e5a15b6fcce83602df58458e0f3b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\textstyle 2}"></span>.</div> </li> <li class="gallerybox" style="width: 335px"> <div class="thumb" style="width: 330px; height: 230px;"><span typeof="mw:File"><a href="/wiki/%D7%A7%D7%95%D7%91%D7%A5:Streckung_der_Summe_zweier_Vektoren.gif" class="mw-file-description" title="הפונקציה היא אדטיבית ושומרת על חיבור: '"`UNIQ--postMath-0000004A-QINU`"'"><img alt="הפונקציה היא אדטיבית ושומרת על חיבור: '"`UNIQ--postMath-0000004A-QINU`"'" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/25/Streckung_der_Summe_zweier_Vektoren.gif/259px-Streckung_der_Summe_zweier_Vektoren.gif" decoding="async" width="259" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/2/25/Streckung_der_Summe_zweier_Vektoren.gif 1.5x" data-file-width="330" data-file-height="255" /></a></span></div> <div class="gallerytext">הפונקציה היא אדטיבית ושומרת על חיבור: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f(a+b)=f(a)+f(b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle f(a+b)=f(a)+f(b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4bd2d7c5d3bfd1b45746f8d695eab2a9ce3b8a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.498ex; height:2.843ex;" alt="{\textstyle f(a+b)=f(a)+f(b)}"></span></div> </li> <li class="gallerybox" style="width: 335px"> <div class="thumb" style="width: 330px; height: 230px;"><span typeof="mw:File"><a href="/wiki/%D7%A7%D7%95%D7%91%D7%A5:Streckung_homogenitaet_Version_3.gif" class="mw-file-description" title="הפונקציה משמרת כפל בסקלר (פונקציה הומוגנית): אפשר למתוח את הקלט ואז להפעיל את הפונקציה או להפעיל את הפונקציה ואז למתוח את הפלט. בניסוח מתמטי: '"`UNIQ--postMath-0000004B-QINU`"'"><img alt="הפונקציה משמרת כפל בסקלר (פונקציה הומוגנית): אפשר למתוח את הקלט ואז להפעיל את הפונקציה או להפעיל את הפונקציה ואז למתוח את הפלט. בניסוח מתמטי: '"`UNIQ--postMath-0000004B-QINU`"'" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Streckung_homogenitaet_Version_3.gif/259px-Streckung_homogenitaet_Version_3.gif" decoding="async" width="259" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/e/e6/Streckung_homogenitaet_Version_3.gif 1.5x" data-file-width="330" data-file-height="255" /></a></span></div> <div class="gallerytext">הפונקציה משמרת כפל בסקלר (<a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94_%D7%94%D7%95%D7%9E%D7%95%D7%92%D7%A0%D7%99%D7%AA" title="פונקציה הומוגנית">פונקציה הומוגנית</a>): אפשר למתוח את הקלט ואז להפעיל את הפונקציה או להפעיל את הפונקציה ואז למתוח את הפלט. בניסוח מתמטי: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f(\lambda a)=\lambda f(a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>λ</mi> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>λ</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle f(\lambda a)=\lambda f(a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79ebb1956d93387884879081217c935f0edae84a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.444ex; height:2.843ex;" alt="{\textstyle f(\lambda a)=\lambda f(a)}"></span></div> </li> </ul> <div class="mw-heading mw-heading2"><h2 id="סוגי_העתקות_ליניאריות"><span id=".D7.A1.D7.95.D7.92.D7.99_.D7.94.D7.A2.D7.AA.D7.A7.D7.95.D7.AA_.D7.9C.D7.99.D7.A0.D7.99.D7.90.D7.A8.D7.99.D7.95.D7.AA"></span>סוגי העתקות ליניאריות</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%94%D7%A2%D7%AA%D7%A7%D7%94_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA&action=edit&section=3" title="עריכת קוד המקור של הפרק: סוגי העתקות ליניאריות"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%94%D7%A2%D7%AA%D7%A7%D7%94_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA&veaction=edit&section=3" title="עריכת פסקה: "סוגי העתקות ליניאריות"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>יהיו <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> ו-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span> מרחבים וקטוריים מעל שדה כלשהו <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span>, ו-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> העתקה ליניארית מ-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> ל-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span>. </p> <ul><li>נאמר ש-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> היא <a href="/wiki/%D7%94%D7%95%D7%9E%D7%95%D7%9E%D7%95%D7%A8%D7%A4%D7%99%D7%96%D7%9D" title="הומומורפיזם">הומומורפיזם</a> (של מרחבים וקטוריים) אם <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> היא העתקה ליניארית.</li> <li>נאמר ש-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> היא מונומורפיזם אם <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> חח"ע (<a href="/wiki/%D7%97%D7%93_%D7%97%D7%93_%D7%A2%D7%A8%D7%9B%D7%99%D7%AA" class="mw-redirect" title="חד חד ערכית">חד חד ערכית</a>).</li> <li>נאמר ש-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> היא אפימורפיזם אם <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> <a href="/wiki/%D7%A2%D7%9C" class="mw-redirect" title="על">על</a>.</li> <li>נאמר ש-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> היא <a href="/wiki/%D7%90%D7%99%D7%96%D7%95%D7%9E%D7%95%D7%A8%D7%A4%D7%99%D7%96%D7%9D" title="איזומורפיזם">איזומורפיזם</a> אם <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> היא הפיכה.</li></ul> <p>מרחבים הם איזומורפיים אם קיימת ביניהם העתקה ליניארית הפיכה, ומתברר כי שני מרחבים וקטוריים איזומורפיים זה לזה אם ורק אם הם שווי-ממד. ה<a href="/wiki/%D7%9E%D7%9E%D7%93_(%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA)" title="ממד (אלגברה ליניארית)">ממד</a> מונוטוני תחת העתקות ליניאריות: אינו יורד תחת העתקה חח"ע, ואינו עולה תחת העתקה על. אם מתקיים שני מרחבים מממד סופי <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> ו-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span> הם שווי ממד, אזי שלוש התכונות לעיל (חח"ע, על, הפיכות) שקולות זו לזו. </p><p>מכאן נובע שכל העתקה על בין מרחב ממד סופי לעצמו היא איזומורפיזם (באופן כללי אובייקט ב<a href="/wiki/%D7%A7%D7%98%D7%92%D7%95%D7%A8%D7%99%D7%94_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="קטגוריה (מתמטיקה)">קטגוריה</a> שמקיים תכונה זאת נקרא <a href="/w/index.php?title=%D7%90%D7%95%D7%91%D7%99%D7%99%D7%A7%D7%98_%D7%94%D7%95%D7%A4%D7%A4%D7%99%D7%90%D7%A0%D7%99&action=edit&redlink=1" class="new" title="אובייקט הופפיאני (הדף אינו קיים)">הופפיאני</a><small class="noprint" dir="rtl">(<a href="https://en.wikipedia.org/wiki/Hopfian_object" class="extiw" title="en:Hopfian object">אנ'</a>)</small>; כך שמרחב ממד סופי הוא הופפיאני). </p><p>באופן דומה נובע שכל העתקה חח"ע בין מרחב ממד סופי לעצמו היא איזומורפיזם (כך שמרחב ממד סופי הוא קו-הופפיאני). </p> <div class="mw-heading mw-heading2"><h2 id="מרחב_ההעתקות_הליניאריות"><span id=".D7.9E.D7.A8.D7.97.D7.91_.D7.94.D7.94.D7.A2.D7.AA.D7.A7.D7.95.D7.AA_.D7.94.D7.9C.D7.99.D7.A0.D7.99.D7.90.D7.A8.D7.99.D7.95.D7.AA"></span>מרחב ההעתקות הליניאריות</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%94%D7%A2%D7%AA%D7%A7%D7%94_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA&action=edit&section=4" title="עריכת קוד המקור של הפרק: מרחב ההעתקות הליניאריות"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%94%D7%A2%D7%AA%D7%A7%D7%94_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA&veaction=edit&section=4" title="עריכת פסקה: "מרחב ההעתקות הליניאריות"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>אוסף כל ההעתקות הליניאריות מ-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span> ל-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a87a024931038d1858dc22e8a194e5978c3412e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.353ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{m}}"></span> מהווה בעצמו מרחב וקטורי מ<a href="/wiki/%D7%9E%D7%9E%D7%93_(%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA)" title="ממד (אלגברה ליניארית)">ממד</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\cdot n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>⋅</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\cdot n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e825052f4b1acf3df49590dacd8612ce5220c19c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.114ex; height:1.676ex;" alt="{\displaystyle m\cdot n}"></span>. על מנת שמשפט זה יהיה מוגדר כהלכה, עלינו להגדיר חיבור של העתקות ליניאריות וכפל בסקלר. את זאת נעשה בדרך הטריוויאלית. אם <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{1},T_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{1},T_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0e6aef110538dac610120d6a38a5e5618810d12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.858ex; height:2.509ex;" alt="{\displaystyle T_{1},T_{2}}"></span> הן העתקות ליניאריות מ-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span> ל-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a87a024931038d1858dc22e8a194e5978c3412e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.353ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{m}}"></span>, ו-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> הוא אבר בשדה אז נגדיר חיבור בין העתקות וכפל של העתקה בסקלר כך: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (T_{1}+T_{2})(v)=T_{1}(v)+T_{2}(v)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (T_{1}+T_{2})(v)=T_{1}(v)+T_{2}(v)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b68c4eb55034a5589d26e0002f7c5bf3357ae72d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.046ex; height:2.843ex;" alt="{\displaystyle (T_{1}+T_{2})(v)=T_{1}(v)+T_{2}(v)}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\alpha T_{1})(v)=\alpha \cdot T_{1}(v)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>α</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>α</mi> <mo>⋅</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\alpha T_{1})(v)=\alpha \cdot T_{1}(v)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c097f5b28390f268ce639c4c858106c42fb8b88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.26ex; height:2.843ex;" alt="{\displaystyle (\alpha T_{1})(v)=\alpha \cdot T_{1}(v)}"></span></li></ul> <p>וקטור האפס הוא טרנספורמציית האפס. </p><p>המרחב הווקטורי של כל ההעתקות ממרחב וקטורי <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> למרחב וקטורי <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span> מסומן <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Hom} (V,W)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Hom</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>V</mi> <mo>,</mo> <mi>W</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Hom} (V,W)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac84f5e6c437b45fbcddb8ae3c3e6f03940e3ed1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.907ex; height:2.843ex;" alt="{\displaystyle \operatorname {Hom} (V,W)}"></span>. אם <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=\mathbb {R} ^{n},\ W=\mathbb {R} ^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>,</mo> <mtext> </mtext> <mi>W</mi> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=\mathbb {R} ^{n},\ W=\mathbb {R} ^{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50c2f6a77a93387c9c73014ae74d92490f9f5ecf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.284ex; height:2.676ex;" alt="{\displaystyle V=\mathbb {R} ^{n},\ W=\mathbb {R} ^{m}}"></span> אז<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Hom} (\mathbb {R} ^{n},\mathbb {R} ^{m})\cong \mathbb {R} ^{m\times n}=M_{m\times n}(\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Hom</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>≅</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Hom} (\mathbb {R} ^{n},\mathbb {R} ^{m})\cong \mathbb {R} ^{m\times n}=M_{m\times n}(\mathbb {R} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccb63d843b1c2125bf937013f08c2cab7c6c4435" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.43ex; height:2.843ex;" alt="{\displaystyle \operatorname {Hom} (\mathbb {R} ^{n},\mathbb {R} ^{m})\cong \mathbb {R} ^{m\times n}=M_{m\times n}(\mathbb {R} )}"></span> כלומר: מרחב הווקטורי של כל ההעתקות הליניאריות <a href="/wiki/%D7%90%D7%99%D7%96%D7%95%D7%9E%D7%95%D7%A8%D7%A4%D7%99%D7%96%D7%9D" title="איזומורפיזם">איזומורפי</a> למרחב המטריצות עם <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> שורות ו-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> עמודות. </p> <div class="mw-heading mw-heading2"><h2 id="גרעין_ותמונה_של_העתקה_ליניארית"><span id=".D7.92.D7.A8.D7.A2.D7.99.D7.9F_.D7.95.D7.AA.D7.9E.D7.95.D7.A0.D7.94_.D7.A9.D7.9C_.D7.94.D7.A2.D7.AA.D7.A7.D7.94_.D7.9C.D7.99.D7.A0.D7.99.D7.90.D7.A8.D7.99.D7.AA"></span>גרעין ותמונה של העתקה ליניארית</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%94%D7%A2%D7%AA%D7%A7%D7%94_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA&action=edit&section=5" title="עריכת קוד המקור של הפרק: גרעין ותמונה של העתקה ליניארית"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%94%D7%A2%D7%AA%D7%A7%D7%94_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA&veaction=edit&section=5" title="עריכת פסקה: "גרעין ותמונה של העתקה ליניארית"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>תהי טרנספורמציה ליניארית <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\colon V\to W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>:</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T\colon V\to W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9307c45fa2557b519d013c5b4229c9bd53a6c3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.507ex; height:2.176ex;" alt="{\displaystyle T\colon V\to W}"></span>. </p><p>ה<a href="/wiki/%D7%92%D7%A8%D7%A2%D7%99%D7%9F_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" class="mw-redirect" title="גרעין (מתמטיקה)">גרעין</a> של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span>, המסומן <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ker(T)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ker</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ker(T)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/295af7fc4717eb9c5eb069fc883753611e57a649" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.617ex; height:2.843ex;" alt="{\displaystyle \ker(T)}"></span> (מהמילה Kernel – גרעין), הוא קבוצת הווקטורים ב-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> המועתקים על ידי <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> לווקטור האפס של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span>. כלומר:<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \ker(T)=\{v\in V\mid T(v)=0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>ker</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>v</mi> <mo>∈</mo> <mi>V</mi> <mo>∣</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \ker(T)=\{v\in V\mid T(v)=0\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51cfe9c4cfcd61c670298b8eb50b6e565d23ed8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.567ex; height:2.843ex;" alt="{\textstyle \ker(T)=\{v\in V\mid T(v)=0\}}"></span>. </p><p>משימוש בתכונות הטרנספורמציה הליניארית קל לראות כי הגרעין הוא <a href="/wiki/%D7%9E%D7%A8%D7%97%D7%91_%D7%95%D7%A7%D7%98%D7%95%D7%A8%D7%99" title="מרחב וקטורי">מרחב וקטורי</a> חלקי (תת-מרחב) ל-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> – משמע, הוא סגור לחיבור וכפל בסקלר. הגרעין <a href="/wiki/%D7%98%D7%A8%D7%99%D7%95%D7%95%D7%99%D7%90%D7%9C%D7%99_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="טריוויאלי (מתמטיקה)">טריוויאלי</a> (דהיינו, שווה לתת-מרחב האפס) בדיוק כאשר <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> חח"ע. </p><p>ה<a href="/wiki/%D7%AA%D7%9E%D7%95%D7%A0%D7%94_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" class="mw-redirect" title="תמונה (מתמטיקה)">תמונה</a> של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span>, המסומנת <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Im} (T)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Im</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Im} (T)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7739d7fc11dbec808c1de4e50280502b53845e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.221ex; height:2.843ex;" alt="{\displaystyle \operatorname {Im} (T)}"></span> (מהמילה Image – תמונה), היא קבוצת איברי <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span> שקיים להם מקור ב-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span>, כלומר: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \operatorname {Im} (T)=\{w\in W\mid \exists v\in V:w=T(v)\}=\{T(v)\mid v\in V\}=T(V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>Im</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>w</mi> <mo>∈</mo> <mi>W</mi> <mo>∣</mo> <mi mathvariant="normal">∃</mi> <mi>v</mi> <mo>∈</mo> <mi>V</mi> <mo>:</mo> <mi>w</mi> <mo>=</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>∣</mo> <mi>v</mi> <mo>∈</mo> <mi>V</mi> <mo fence="false" stretchy="false">}</mo> <mo>=</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \operatorname {Im} (T)=\{w\in W\mid \exists v\in V:w=T(v)\}=\{T(v)\mid v\in V\}=T(V)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f06568767da9ddb66b6cb4084bfcf0b710e9701" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:65.831ex; height:2.843ex;" alt="{\displaystyle \ \operatorname {Im} (T)=\{w\in W\mid \exists v\in V:w=T(v)\}=\{T(v)\mid v\in V\}=T(V)}"></span> גם התמונה של טרנספורמציה ליניארית סגורה לחיבור וכפל בסקלר, ולכן מהווה תת-<a href="/wiki/%D7%9E%D7%A8%D7%97%D7%91_%D7%95%D7%A7%D7%98%D7%95%D7%A8%D7%99" title="מרחב וקטורי">מרחב וקטורי</a> של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="משפט_הדרגה"><span id=".D7.9E.D7.A9.D7.A4.D7.98_.D7.94.D7.93.D7.A8.D7.92.D7.94"></span>משפט הדרגה</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%94%D7%A2%D7%AA%D7%A7%D7%94_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA&action=edit&section=6" title="עריכת קוד המקור של הפרק: משפט הדרגה"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%94%D7%A2%D7%AA%D7%A7%D7%94_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA&veaction=edit&section=6" title="עריכת פסקה: "משפט הדרגה"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>משפט הדרגה</b>: לכל מרחב <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> מממד סופי ולכל טרנספורמציה ליניארית <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T:V\rightarrow W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>:</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T:V\rightarrow W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50d25ad658e5e0cc92361d12eaf1b6a1904fb233" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.41ex; height:2.176ex;" alt="{\displaystyle T:V\rightarrow W}"></span> מתקיים:</p><div style="text-align: center;"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \dim(\operatorname {Im} (T))+\dim(\ker(T))=\dim(V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>Im</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>+</mo> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ker</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dim(\operatorname {Im} (T))+\dim(\ker(T))=\dim(V)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/397a337f8ddd1ef74957882c246657d143ba126a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.618ex; height:2.843ex;" alt="{\displaystyle \dim(\operatorname {Im} (T))+\dim(\ker(T))=\dim(V)}"></span>. </p> </div> <p>נשים לב כי בנוסחה אין תלות כלל בממד של הטווח <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span>, אלא רק בממד של התחום <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span>, ה<a href="/wiki/%D7%92%D7%A8%D7%A2%D7%99%D7%9F_(%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94)" title="גרעין (אלגברה)">גרעין</a> ו<a href="/wiki/%D7%AA%D7%9E%D7%95%D7%A0%D7%94_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" class="mw-redirect" title="תמונה (מתמטיקה)">התמונה</a>. נובעים מן המשפט האי-שוויונות הבאים: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \dim(\operatorname {Im} (T))\leq \dim(W)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>Im</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>W</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dim(\operatorname {Im} (T))\leq \dim(W)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54131f0936b430c24198a78f29a55c23b08433b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.124ex; height:2.843ex;" alt="{\displaystyle \dim(\operatorname {Im} (T))\leq \dim(W)}"></span> ולכן <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \dim(\ker(T))\geq \dim(V)-\dim(W)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ker</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>≥</mo> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>W</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dim(\ker(T))\geq \dim(V)-\dim(W)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dad8c92daaf9c5645421d337a01ed635441b333c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.832ex; height:2.843ex;" alt="{\displaystyle \dim(\ker(T))\geq \dim(V)-\dim(W)}"></span> </p><p><br /> </p> <table class="wikitable mw-collapsible" style="width: 100%; margin: auto;"> <tbody><tr> <th style="">הוכחה </th></tr> <tr> <td>יהיו <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\ v_{1},...,\ v_{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mtext> </mtext> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mtext> </mtext> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\ v_{1},...,\ v_{n}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2a51a02deac58377987e853dac0d902781a85bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.184ex; height:2.843ex;" alt="{\displaystyle \{\ v_{1},...,\ v_{n}\}}"></span> <a href="/wiki/%D7%91%D7%A1%D7%99%D7%A1_(%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94)" title="בסיס (אלגברה)">בסיס</a> של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ker(T)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ker</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ker(T)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/295af7fc4717eb9c5eb069fc883753611e57a649" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.617ex; height:2.843ex;" alt="{\displaystyle \ker(T)}"></span> ויהיו <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\ u_{1},...,\ u_{m}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mtext> </mtext> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mtext> </mtext> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\ u_{1},...,\ u_{m}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2442450d59da798a4be9fb1da626d9e2d9828dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.045ex; height:2.843ex;" alt="{\displaystyle \{\ u_{1},...,\ u_{m}\}}"></span> וקטורים כך ש-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\ T(u_{1}),...,\ T(u_{m})\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mtext> </mtext> <mi>T</mi> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mtext> </mtext> <mi>T</mi> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\ T(u_{1}),...,\ T(u_{m})\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f4a61aeb9297fc62c219b101af29777497999f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.936ex; height:2.843ex;" alt="{\displaystyle \{\ T(u_{1}),...,\ T(u_{m})\}}"></span> מהווים בסיס ל-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Im} (T)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Im</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Im} (T)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7739d7fc11dbec808c1de4e50280502b53845e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.221ex; height:2.843ex;" alt="{\displaystyle \operatorname {Im} (T)}"></span> <p>צ"ל: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\ v_{1},...,\ v_{n},\ u_{1},...,\ u_{m}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mtext> </mtext> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mtext> </mtext> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mtext> </mtext> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\ v_{1},...,\ v_{n},\ u_{1},...,\ u_{m}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4db987a107af4b5eeaafe5bc3d6bf3b3485bea6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.937ex; height:2.843ex;" alt="{\displaystyle \{\ v_{1},...,\ v_{n},\ u_{1},...,\ u_{m}\}}"></span> מהווה בסיס ל-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span>. </p><p>נראה כי הקבוצה <a href="/wiki/%D7%AA%D7%9C%D7%95%D7%AA_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA" title="תלות ליניארית">בת"ל</a> (בלתי תלויה ליניארית): יהיו <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1},...,\ a_{n},\ b_{1},...,\ b_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mtext> </mtext> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mtext> </mtext> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1},...,\ a_{n},\ b_{1},...,\ b_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69691a889b9dec651cb7be42b498d575abe73f28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.572ex; height:2.509ex;" alt="{\displaystyle a_{1},...,\ a_{n},\ b_{1},...,\ b_{m}}"></span> סקלרים כך ש- <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{n}{a_{i}v_{i}}+\sum _{j=1}^{m}{b_{j}u_{j}}=0_{V}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>=</mo> <msub> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n}{a_{i}v_{i}}+\sum _{j=1}^{m}{b_{j}u_{j}}=0_{V}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77696c50c8cc63d802c05ee3c1310e8d2d820a36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:24.185ex; height:7.176ex;" alt="{\displaystyle \sum _{i=1}^{n}{a_{i}v_{i}}+\sum _{j=1}^{m}{b_{j}u_{j}}=0_{V}}"></span> </p><p>לכן בהכרח <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0_{W}=\ T(0_{V})=\ T(\sum _{i=1}^{n}{a_{i}v_{i}}+\sum _{j=1}^{m}{b_{j}u_{j}})=\sum _{i=1}^{n}{a_{i}\ T(v_{i})}+\sum _{j=1}^{m}{b_{j}\ T(u_{j})}=0+\sum _{j=1}^{m}{b_{j}\ T(u_{j})}=\sum _{j=1}^{m}{b_{j}\ T(u_{j})}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>W</mi> </mrow> </msub> <mo>=</mo> <mtext> </mtext> <mi>T</mi> <mo stretchy="false">(</mo> <msub> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mtext> </mtext> <mi>T</mi> <mo stretchy="false">(</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mtext> </mtext> <mi>T</mi> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mtext> </mtext> <mi>T</mi> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>0</mn> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mtext> </mtext> <mi>T</mi> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mtext> </mtext> <mi>T</mi> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0_{W}=\ T(0_{V})=\ T(\sum _{i=1}^{n}{a_{i}v_{i}}+\sum _{j=1}^{m}{b_{j}u_{j}})=\sum _{i=1}^{n}{a_{i}\ T(v_{i})}+\sum _{j=1}^{m}{b_{j}\ T(u_{j})}=0+\sum _{j=1}^{m}{b_{j}\ T(u_{j})}=\sum _{j=1}^{m}{b_{j}\ T(u_{j})}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64d53d9886b0fa1234433d1c747ec62393cd919b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:102.061ex; height:7.176ex;" alt="{\displaystyle 0_{W}=\ T(0_{V})=\ T(\sum _{i=1}^{n}{a_{i}v_{i}}+\sum _{j=1}^{m}{b_{j}u_{j}})=\sum _{i=1}^{n}{a_{i}\ T(v_{i})}+\sum _{j=1}^{m}{b_{j}\ T(u_{j})}=0+\sum _{j=1}^{m}{b_{j}\ T(u_{j})}=\sum _{j=1}^{m}{b_{j}\ T(u_{j})}}"></span> </p><p>לכן, כיוון ש-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\ T(u_{1}),...,\ T(u_{m})\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mtext> </mtext> <mi>T</mi> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mtext> </mtext> <mi>T</mi> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\ T(u_{1}),...,\ T(u_{m})\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f4a61aeb9297fc62c219b101af29777497999f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.936ex; height:2.843ex;" alt="{\displaystyle \{\ T(u_{1}),...,\ T(u_{m})\}}"></span> בת"ל, לכל <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}"></span> מתקיים <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{j}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{j}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f422b1b38dfb3c5a6e5b5e8abe33f68e153213db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.168ex; height:2.843ex;" alt="{\displaystyle b_{j}=0}"></span> </p><p>נציב ב-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{n}{a_{i}v_{i}}+\sum _{j=1}^{m}{b_{j}u_{j}}=0_{V}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>=</mo> <msub> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n}{a_{i}v_{i}}+\sum _{j=1}^{m}{b_{j}u_{j}}=0_{V}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77696c50c8cc63d802c05ee3c1310e8d2d820a36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:24.185ex; height:7.176ex;" alt="{\displaystyle \sum _{i=1}^{n}{a_{i}v_{i}}+\sum _{j=1}^{m}{b_{j}u_{j}}=0_{V}}"></span> ונקבל כי <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0_{V}=\sum _{i=1}^{n}{a_{i}v_{i}}+\sum _{j=1}^{m}{b_{j}u_{j}}=\sum _{i=1}^{n}{a_{i}v_{i}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0_{V}=\sum _{i=1}^{n}{a_{i}v_{i}}+\sum _{j=1}^{m}{b_{j}u_{j}}=\sum _{i=1}^{n}{a_{i}v_{i}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/230b7f63afd4383a0fbc64ff415981c185c6ffc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:34.982ex; height:7.176ex;" alt="{\displaystyle 0_{V}=\sum _{i=1}^{n}{a_{i}v_{i}}+\sum _{j=1}^{m}{b_{j}u_{j}}=\sum _{i=1}^{n}{a_{i}v_{i}}}"></span> </p><p>וכיוון ש-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\ v_{1},...,\ v_{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mtext> </mtext> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mtext> </mtext> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\ v_{1},...,\ v_{n}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2a51a02deac58377987e853dac0d902781a85bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.184ex; height:2.843ex;" alt="{\displaystyle \{\ v_{1},...,\ v_{n}\}}"></span> בת"ל נקבל כי לכל <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> מתקיים כי <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1d2e283d81c74e9c6742dcbbcad8c622ef0c3c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.29ex; height:2.509ex;" alt="{\displaystyle a_{i}=0}"></span> </p><p>כלומר, רק עבור הסקלרים הטריביאליים ה<a href="/wiki/%D7%A7%D7%95%D7%9E%D7%91%D7%99%D7%A0%D7%A6%D7%99%D7%94" class="mw-redirect" title="קומבינציה">קומבינציה</a> של הקבוצה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\ v_{1},...,\ v_{n},\ u_{1},...,\ u_{m}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mtext> </mtext> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mtext> </mtext> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mtext> </mtext> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\ v_{1},...,\ v_{n},\ u_{1},...,\ u_{m}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4db987a107af4b5eeaafe5bc3d6bf3b3485bea6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.937ex; height:2.843ex;" alt="{\displaystyle \{\ v_{1},...,\ v_{n},\ u_{1},...,\ u_{m}\}}"></span> שווה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0_{V}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0_{V}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a56ff4d32c0398c1f7d139b1107fbd654d078fb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.658ex; height:2.509ex;" alt="{\displaystyle 0_{V}}"></span> ולכן הקבוצה בת”ל מעל <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span>. </p><p>נראה כי הקבוצה <a href="/wiki/%D7%A7%D7%91%D7%95%D7%A6%D7%94_%D7%A4%D7%95%D7%A8%D7%A9%D7%AA" title="קבוצה פורשת">פורשת</a>: יהי <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}\in \ V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>∈</mo> <mtext> </mtext> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}\in \ V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef37c3f59222ee969d283f22d7fb75721d021698" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.384ex; height:2.343ex;" alt="{\displaystyle {\vec {v}}\in \ V}"></span>, מתקיים <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T({\vec {v}})\in \ \operatorname {Im} (T)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>∈</mo> <mtext> </mtext> <mi>Im</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T({\vec {v}})\in \ \operatorname {Im} (T)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cf1f7eec40633ed2f80b4b76b328cf36101dc59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.65ex; height:2.843ex;" alt="{\displaystyle T({\vec {v}})\in \ \operatorname {Im} (T)}"></span> לכן, קיימים סקלרים <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{1},...,\ b_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mtext> </mtext> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{1},...,\ b_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/220d23aedb2f917fa929b3fdaaaca0ae6b70ed07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.475ex; height:2.509ex;" alt="{\displaystyle b_{1},...,\ b_{m}}"></span> כך ש-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{j=1}^{m}{b_{j}T(u_{j})}=\ T({\vec {v}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mi>T</mi> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mtext> </mtext> <mi>T</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{j=1}^{m}{b_{j}T(u_{j})}=\ T({\vec {v}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d616e33a9d635b17b0af5c0324fe6403d763fa5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:19.634ex; height:7.176ex;" alt="{\displaystyle \sum _{j=1}^{m}{b_{j}T(u_{j})}=\ T({\vec {v}})}"></span>. </p><p>נעביר אגפים ונקבל כי <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0_{W}=\ T({\vec {v}})-\sum _{j=1}^{m}{b_{j}T(u_{j})}=\ T({\vec {v}}-\sum _{j=1}^{m}{b_{j}u_{j}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>W</mi> </mrow> </msub> <mo>=</mo> <mtext> </mtext> <mi>T</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mi>T</mi> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mtext> </mtext> <mi>T</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0_{W}=\ T({\vec {v}})-\sum _{j=1}^{m}{b_{j}T(u_{j})}=\ T({\vec {v}}-\sum _{j=1}^{m}{b_{j}u_{j}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bda881c22ee53efbe46fbb9ca97996817c1caed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:44.621ex; height:7.176ex;" alt="{\displaystyle 0_{W}=\ T({\vec {v}})-\sum _{j=1}^{m}{b_{j}T(u_{j})}=\ T({\vec {v}}-\sum _{j=1}^{m}{b_{j}u_{j}})}"></span> ולכן <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}-\sum _{j=1}^{m}{b_{j}u_{j}}\in \ \ker(T)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>∈</mo> <mtext> </mtext> <mi>ker</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}-\sum _{j=1}^{m}{b_{j}u_{j}}\in \ \ker(T)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/898a6934f8044710368c7269099b937ee9575b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:22.33ex; height:7.176ex;" alt="{\displaystyle {\vec {v}}-\sum _{j=1}^{m}{b_{j}u_{j}}\in \ \ker(T)}"></span> </p><p>לכן, קיימים סקלרים <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1},...,\ a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mtext> </mtext> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1},...,\ a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9fa38712d534d6a6fa447f6efbae93d23be33a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.483ex; height:2.009ex;" alt="{\displaystyle a_{1},...,\ a_{n}}"></span> כך ש-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}-\sum _{j=1}^{m}{b_{j}u_{j}}=\sum _{i=1}^{n}{a_{i}v_{i}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}-\sum _{j=1}^{m}{b_{j}u_{j}}=\sum _{i=1}^{n}{a_{i}v_{i}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76edbad82a2febb09b77d41b8b59dddd1cdf3acf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:22.702ex; height:7.176ex;" alt="{\displaystyle {\vec {v}}-\sum _{j=1}^{m}{b_{j}u_{j}}=\sum _{i=1}^{n}{a_{i}v_{i}}}"></span> ולכן <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}=\sum _{i=1}^{n}{a_{i}v_{i}}+\sum _{j=1}^{m}{b_{j}u_{j}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}=\sum _{i=1}^{n}{a_{i}v_{i}}+\sum _{j=1}^{m}{b_{j}u_{j}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04c61295499bcd4a9594378a3b36f63d139f24ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:22.702ex; height:7.176ex;" alt="{\displaystyle {\vec {v}}=\sum _{i=1}^{n}{a_{i}v_{i}}+\sum _{j=1}^{m}{b_{j}u_{j}}}"></span> </p><p>ולכן הקבוצה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\ v_{1},...,\ v_{n},\ u_{1},...,\ u_{m}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mtext> </mtext> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mtext> </mtext> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mtext> </mtext> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\ v_{1},...,\ v_{n},\ u_{1},...,\ u_{m}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4db987a107af4b5eeaafe5bc3d6bf3b3485bea6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.937ex; height:2.843ex;" alt="{\displaystyle \{\ v_{1},...,\ v_{n},\ u_{1},...,\ u_{m}\}}"></span> פורשת את <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span>. </p><p>לכן הקבוצה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\ v_{1},...,\ v_{n},\ u_{1},...,\ u_{m}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mtext> </mtext> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mtext> </mtext> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mtext> </mtext> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\ v_{1},...,\ v_{n},\ u_{1},...,\ u_{m}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4db987a107af4b5eeaafe5bc3d6bf3b3485bea6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.937ex; height:2.843ex;" alt="{\displaystyle \{\ v_{1},...,\ v_{n},\ u_{1},...,\ u_{m}\}}"></span> מהווה בסיס עבור <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> ולכן מתקיים <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \dim(\operatorname {Im} (T))+\dim(\ker(T))=\ n+\ m=\dim(V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>Im</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>+</mo> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ker</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mtext> </mtext> <mi>n</mi> <mo>+</mo> <mtext> </mtext> <mi>m</mi> <mo>=</mo> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dim(\operatorname {Im} (T))+\dim(\ker(T))=\ n+\ m=\dim(V)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/546da44a22d944a3c515fd60af8cacf0e9a71520" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.153ex; height:2.843ex;" alt="{\displaystyle \dim(\operatorname {Im} (T))+\dim(\ker(T))=\ n+\ m=\dim(V)}"></span>. <b>מ.ש.ל.</b> </p> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="מטריצה_מייצגת_של_העתקה_ליניארית"><span id=".D7.9E.D7.98.D7.A8.D7.99.D7.A6.D7.94_.D7.9E.D7.99.D7.99.D7.A6.D7.92.D7.AA_.D7.A9.D7.9C_.D7.94.D7.A2.D7.AA.D7.A7.D7.94_.D7.9C.D7.99.D7.A0.D7.99.D7.90.D7.A8.D7.99.D7.AA"></span>מטריצה מייצגת של העתקה ליניארית</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%94%D7%A2%D7%AA%D7%A7%D7%94_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA&action=edit&section=7" title="עריכת קוד המקור של הפרק: מטריצה מייצגת של העתקה ליניארית"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%94%D7%A2%D7%AA%D7%A7%D7%94_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA&veaction=edit&section=7" title="עריכת פסקה: "מטריצה מייצגת של העתקה ליניארית"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>יהיו <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> ו-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span> מרחבים וקטורים מממד סופי מעל שדה כלשהו <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span>, ו-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\colon V\to W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>:</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T\colon V\to W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9307c45fa2557b519d013c5b4229c9bd53a6c3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.507ex; height:2.176ex;" alt="{\displaystyle T\colon V\to W}"></span> העתקה ליניארית מ-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> ל-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span>. נקבע <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B=\{v_{1},...,v_{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B=\{v_{1},...,v_{n}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a42b541b9dd200a5fe63cec50bc550ef28abd8b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.885ex; height:2.843ex;" alt="{\displaystyle B=\{v_{1},...,v_{n}\}}"></span> בסיס של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> ו-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C=\{w_{1},...,w_{m}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C=\{w_{1},...,w_{m}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdde1cbe05b173d436dc551e54b53faf426f769e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.417ex; height:2.843ex;" alt="{\displaystyle C=\{w_{1},...,w_{m}\}}"></span> בסיס של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span>. לכל וקטור ב-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> ניתן להתאים את <a href="/wiki/%D7%95%D7%A7%D7%98%D7%95%D7%A8_%D7%A7%D7%95%D7%90%D7%95%D7%A8%D7%93%D7%99%D7%A0%D7%98%D7%95%D7%AA" class="mw-redirect" title="וקטור קואורדינטות">וקטור הקואורדינטות</a> שלו לפי הבסיס <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span>, ‏<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\ni v\mapsto [v]_{B}\in \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>∋</mo> <mi>v</mi> <mo stretchy="false">↦</mo> <mo stretchy="false">[</mo> <mi>v</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V\ni v\mapsto [v]_{B}\in \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c7e8a612f6e6ebe0ca4d37333d1453530cb2d16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.008ex; height:2.843ex;" alt="{\displaystyle V\ni v\mapsto [v]_{B}\in \mathbb {R} ^{n}}"></span> ובאופן דומה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W\ni w\mapsto [w]_{C}\in \mathbb {R} ^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>∋</mo> <mi>w</mi> <mo stretchy="false">↦</mo> <mo stretchy="false">[</mo> <mi>w</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W\ni w\mapsto [w]_{C}\in \mathbb {R} ^{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f572c6e5f5dedf3d54c3247b0dd8af2141de7253" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.187ex; height:2.843ex;" alt="{\displaystyle W\ni w\mapsto [w]_{C}\in \mathbb {R} ^{m}}"></span>. </p><p>נגדיר את <a href="/wiki/%D7%9E%D7%98%D7%A8%D7%99%D7%A6%D7%94_%D7%9E%D7%99%D7%99%D7%A6%D7%92%D7%AA" class="mw-redirect" title="מטריצה מייצגת">המטריצה המייצגת</a> של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> ביחס לבסיסים <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> ו-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span>, שתסומן <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {[T]_{C}^{B}}\in F^{m\times n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mi>T</mi> <msubsup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msubsup> </mrow> <mo>∈</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {[T]_{C}^{B}}\in F^{m\times n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3e021e51cf189896215f6bebb9c352932490bdb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.006ex; height:3.176ex;" alt="{\displaystyle {[T]_{C}^{B}}\in F^{m\times n}}"></span> כמטריצה שמקיימת את הקשר הבא: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall v\in V:\ \ [T(v)]_{C}=[T]_{C}^{B}[v]_{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>v</mi> <mo>∈</mo> <mi>V</mi> <mo>:</mo> <mtext> </mtext> <mtext> </mtext> <mo stretchy="false">[</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">[</mo> <mi>T</mi> <msubsup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msubsup> <mo stretchy="false">[</mo> <mi>v</mi> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall v\in V:\ \ [T(v)]_{C}=[T]_{C}^{B}[v]_{B}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/501299b88d43a28d76f037f777a6fcf865dc35be" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.905ex; height:3.176ex;" alt="{\displaystyle \forall v\in V:\ \ [T(v)]_{C}=[T]_{C}^{B}[v]_{B}}"></span> ניתן לחשב אותה במפורש על ידי הנוסחה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({[T]_{C}^{B}}\right)_{ij}:=([T(v_{j})]_{C})_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mi>T</mi> <msubsup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>:=</mo> <mo stretchy="false">(</mo> <mo stretchy="false">[</mo> <mi>T</mi> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({[T]_{C}^{B}}\right)_{ij}:=([T(v_{j})]_{C})_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/141d739e9ae4455d5f81ecea304f170b2c4e5ba4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:22.63ex; height:3.676ex;" alt="{\displaystyle \left({[T]_{C}^{B}}\right)_{ij}:=([T(v_{j})]_{C})_{i}}"></span> אך בפועל ישנן דרכים פשוטות יותר לחשב אותה. </p> <div class="mw-heading mw-heading3"><h3 id="תכונות"><span id=".D7.AA.D7.9B.D7.95.D7.A0.D7.95.D7.AA"></span>תכונות</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%94%D7%A2%D7%AA%D7%A7%D7%94_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA&action=edit&section=8" title="עריכת קוד המקור של הפרק: תכונות"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%94%D7%A2%D7%AA%D7%A7%D7%94_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA&veaction=edit&section=8" title="עריכת פסקה: "תכונות"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>נסמן ב-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>B</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b144e466a3757b4706daf24a45d4f1266700016" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.449ex; height:2.509ex;" alt="{\displaystyle B'}"></span> בסיס נוסף של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> וב-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3f732043ebc78ca911b9d7801fd8787b1291b2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.483ex; height:2.509ex;" alt="{\displaystyle C'}"></span> בסיס נוסף של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span>. נסמן ב-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{B'}^{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>B</mi> <mo>′</mo> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{B'}^{B}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3727a447ffe502e3e324b5c898a31aed347c43ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.265ex; height:3.343ex;" alt="{\displaystyle M_{B'}^{B}}"></span>את <a href="/wiki/%D7%9E%D7%98%D7%A8%D7%99%D7%A6%D7%AA_%D7%9E%D7%A2%D7%91%D7%A8" title="מטריצת מעבר">מטריצת המעבר</a> מבסיס <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> לבסיס <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>B</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b144e466a3757b4706daf24a45d4f1266700016" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.449ex; height:2.509ex;" alt="{\displaystyle B'}"></span> וב-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{C'}^{C}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>C</mi> <mo>′</mo> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{C'}^{C}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f4afe7345f8c4b8ce0df3468e6a6ebe59f27a23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:4.289ex; height:3.509ex;" alt="{\displaystyle M_{C'}^{C}}"></span> את מטריצת המעבר מבסיס <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> לבסיס <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3f732043ebc78ca911b9d7801fd8787b1291b2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.483ex; height:2.509ex;" alt="{\displaystyle C'}"></span>. אזי <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [T]_{C'}^{B'}=M_{C'}^{C}[T]_{C}^{B}M_{B}^{B'}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>T</mi> <msubsup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>C</mi> <mo>′</mo> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>B</mi> <mo>′</mo> </msup> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>C</mi> <mo>′</mo> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msubsup> <mo stretchy="false">[</mo> <mi>T</mi> <msubsup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msubsup> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>B</mi> <mo>′</mo> </msup> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [T]_{C'}^{B'}=M_{C'}^{C}[T]_{C}^{B}M_{B}^{B'}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c394050b44470c4c784122762cd38e1f83bcd03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:21.274ex; height:3.676ex;" alt="{\displaystyle [T]_{C'}^{B'}=M_{C'}^{C}[T]_{C}^{B}M_{B}^{B'}}"></span> כאשר <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{B}^{B'}=\left(M_{B'}^{B}\right)^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>B</mi> <mo>′</mo> </msup> </mrow> </msubsup> <mo>=</mo> <msup> <mrow> <mo>(</mo> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>B</mi> <mo>′</mo> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msubsup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{B}^{B'}=\left(M_{B'}^{B}\right)^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2954f3a1ffc39a2a31caee807e58d00225f3cf6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:16.336ex; height:3.843ex;" alt="{\displaystyle M_{B}^{B'}=\left(M_{B'}^{B}\right)^{-1}}"></span>.</li> <li>יהי <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span> מרחב וקטורי נוסף מעל <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> עם בסיס <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span> ותהי <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S:W\rightarrow U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>:</mo> <mi>W</mi> <mo stretchy="false">→</mo> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S:W\rightarrow U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e92053150e5ddb76a4001842bd250645affac01c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.268ex; height:2.176ex;" alt="{\displaystyle S:W\rightarrow U}"></span> העתקה ליניארית נוספת. אזי <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [S\circ T]_{D}^{B}=[S]_{D}^{C}[T]_{C}^{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>S</mi> <mo>∘</mo> <mi>T</mi> <msubsup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msubsup> <mo>=</mo> <mo stretchy="false">[</mo> <mi>S</mi> <msubsup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msubsup> <mo stretchy="false">[</mo> <mi>T</mi> <msubsup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [S\circ T]_{D}^{B}=[S]_{D}^{C}[T]_{C}^{B}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e888e5a6bbb7f8a8f315db47e0508e7a1fcee66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.112ex; height:3.176ex;" alt="{\displaystyle [S\circ T]_{D}^{B}=[S]_{D}^{C}[T]_{C}^{B}}"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="ההעתקה_הדואלית"><span id=".D7.94.D7.94.D7.A2.D7.AA.D7.A7.D7.94_.D7.94.D7.93.D7.95.D7.90.D7.9C.D7.99.D7.AA"></span>ההעתקה הדואלית</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%94%D7%A2%D7%AA%D7%A7%D7%94_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA&action=edit&section=9" title="עריכת קוד המקור של הפרק: ההעתקה הדואלית"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%94%D7%A2%D7%AA%D7%A7%D7%94_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA&veaction=edit&section=9" title="עריכת פסקה: "ההעתקה הדואלית"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>יהיו <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> ו-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span> מרחבים וקטורים מעל שדה כלשהו <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span>, ו-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> העתקה ליניארית מ-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> ל-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span>. </p><p>נתבונן ב<a href="/wiki/%D7%9E%D7%A8%D7%97%D7%91_%D7%93%D7%95%D7%90%D7%9C%D7%99" title="מרחב דואלי">מרחבים הדואליים <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5910e6a94f4f7ee2ee85ceed9dacef3eff7a6242" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle V^{*}}"></span> ו-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5292cb3dbf333bb0b51f39fb9f724fa8d4dfcc63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.562ex; height:2.343ex;" alt="{\displaystyle W^{*}}"></span></a> ונגדיר את ההעתקה הדואלית <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T^{*}:W^{*}\to V^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>:</mo> <msup> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T^{*}:W^{*}\to V^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a1bc9801bac47204cedb8674ec8bbf0381da58d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.859ex; height:2.343ex;" alt="{\displaystyle T^{*}:W^{*}\to V^{*}}"></span> על ידי: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (T^{*}(\varphi ))(v):=\varphi (T(v))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (T^{*}(\varphi ))(v):=\varphi (T(v))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbf5f4427c67f7adaafd739de32b5353c163a486" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.498ex; height:2.843ex;" alt="{\displaystyle (T^{*}(\varphi ))(v):=\varphi (T(v))}"></span> </p><p>כאשר <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \in W^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>∈</mo> <msup> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi \in W^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73691c7a05c10a7beb3ea458bced53a919299cfb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.923ex; height:2.843ex;" alt="{\displaystyle \varphi \in W^{*}}"></span> ו-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\in V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>∈</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v\in V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99886ebbde63daa0224fb9bf56fa11b3c8a6f4fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.756ex; height:2.176ex;" alt="{\displaystyle v\in V}"></span>. עבור תת-מרחב <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{0}\leq V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>≤</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{0}\leq V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53b57cb768608c3a230388d96960d28e73e4596d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.295ex; height:2.509ex;" alt="{\displaystyle V_{0}\leq V}"></span> נסמן: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{0}^{\circ }=\{\varphi \in V^{*}|\varphi (V_{0})=0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msubsup> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>φ</mi> <mo>∈</mo> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{0}^{\circ }=\{\varphi \in V^{*}|\varphi (V_{0})=0\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2250cc7bd86a568d8b72d32095c7f6698da96602" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.373ex; height:3.009ex;" alt="{\displaystyle V_{0}^{\circ }=\{\varphi \in V^{*}|\varphi (V_{0})=0\}}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="תכונות_2"><span id=".D7.AA.D7.9B.D7.95.D7.A0.D7.95.D7.AA_2"></span>תכונות</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%94%D7%A2%D7%AA%D7%A7%D7%94_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA&action=edit&section=10" title="עריכת קוד המקור של הפרק: תכונות"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%94%D7%A2%D7%AA%D7%A7%D7%94_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA&veaction=edit&section=10" title="עריכת פסקה: "תכונות"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5426ed7abdea7d2ff995ad0e01bc4ca62a273855" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.774ex; height:2.343ex;" alt="{\displaystyle T^{*}}"></span> היא העתקה ליניארית ומוגדרת היטב.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Ker}}(T^{*})=({\text{Im}}\ T)^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Ker</mtext> </mrow> <mo stretchy="false">(</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Im</mtext> </mrow> <mtext> </mtext> <mi>T</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{Ker}}(T^{*})=({\text{Im}}\ T)^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2b23e4f27cc4eab5b82999ef569bb75bfca4205" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.29ex; height:2.843ex;" alt="{\displaystyle {\text{Ker}}(T^{*})=({\text{Im}}\ T)^{\circ }}"></span>.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Im} (T^{*})=({\text{Ker}}\ T)^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Im</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Ker</mtext> </mrow> <mtext> </mtext> <mi>T</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Im} (T^{*})=({\text{Ker}}\ T)^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2be341ce332dc852bfc68d77056293d69c08ef9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.29ex; height:2.843ex;" alt="{\displaystyle \operatorname {Im} (T^{*})=({\text{Ker}}\ T)^{\circ }}"></span>.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> חח"ע אם ורק אם <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5426ed7abdea7d2ff995ad0e01bc4ca62a273855" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.774ex; height:2.343ex;" alt="{\displaystyle T^{*}}"></span> על.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5426ed7abdea7d2ff995ad0e01bc4ca62a273855" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.774ex; height:2.343ex;" alt="{\displaystyle T^{*}}"></span> חח"ע אם ורק אם <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> על.</li> <li>יהי <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> בסיס ל-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> ו-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> בסיס ל-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span>. נסמן ב-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202be1fdef03f14a352508cf0bc32073470d44ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.818ex; height:2.343ex;" alt="{\displaystyle B^{*}}"></span> וב-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fda87fa9eddc6a89e202bdebaa9a5e1a55dec9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.852ex; height:2.343ex;" alt="{\displaystyle C^{*}}"></span> בסיסים דואליים ל-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> ו-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> בהתאמה. אזי: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {[T^{*}]_{B^{*}}^{C^{*}}}=({[T^{*}]_{C}^{B}})^{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <msubsup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mrow> </msubsup> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <msubsup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msubsup> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {[T^{*}]_{B^{*}}^{C^{*}}}=({[T^{*}]_{C}^{B}})^{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc356af4840f37dc4af912f50339f04ff1595369" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:17.686ex; height:3.509ex;" alt="{\displaystyle {[T^{*}]_{B^{*}}^{C^{*}}}=({[T^{*}]_{C}^{B}})^{t}}"></span>.</li> <li>נסמן ב-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{V}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{V}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00c18988639f2937d3fe4dc00dfa0b0bd7e0bec1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.158ex; height:2.509ex;" alt="{\displaystyle C_{V}}"></span> וב-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{W}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>W</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{W}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b55f0a36f09f4dd96b4ada2d125df407f45554e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.616ex; height:2.509ex;" alt="{\displaystyle C_{W}}"></span> את <a rel="nofollow" class="external text" href="https://math.stackexchange.com/questions/51283/canonical-isomorphism-between-vector-spaces">האיזומורפיזמים הקנוניים</a> של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> ו-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span> בהתאמה. נגדיר: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T^{**}:=(T^{*})^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> <mo>∗</mo> </mrow> </msup> <mo>:=</mo> <mo stretchy="false">(</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T^{**}:=(T^{*})^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9994b147e50483a4326df249558ec55e11e5bbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.979ex; height:2.843ex;" alt="{\displaystyle T^{**}:=(T^{*})^{*}}"></span>. אז מתקיים: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T^{**}\circ C_{V}=C_{W}\circ T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> <mo>∗</mo> </mrow> </msup> <mo>∘</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>W</mi> </mrow> </msub> <mo>∘</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T^{**}\circ C_{V}=C_{W}\circ T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fade893bbef86370cb18659a12528504448d467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.494ex; height:2.676ex;" alt="{\displaystyle T^{**}\circ C_{V}=C_{W}\circ T}"></span>.</li></ul> <p><br /> </p> <table class="navbox nowraplinks mw-collapsible autocollapse" style="width: 90%; clear: both; margin: 0.5em auto; margin-top: 0.5em; margin-bottom: 0.5em; padding: 0.2em; text-align: right;"> <tbody><tr> <th colspan="3" style="text-align: center; padding-top: 0.1em; padding-bottom: 0.1em; color: black; background:#d1eeee; font-weight: bold;">נושאים ב<a href="/wiki/%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA" title="אלגברה ליניארית">אלגברה ליניארית</a> </th></tr> <tr> <td style="background-color: #F2F3F4; text-align: right; font-weight: bold; padding-left: 5px;">מושגי יסוד </td> <td style="padding-right: 5px; text-align: right;"><a href="/wiki/%D7%A9%D7%93%D7%94_(%D7%9E%D7%91%D7%A0%D7%94_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99)" title="שדה (מבנה אלגברי)">שדה</a> • <a href="/wiki/%D7%9E%D7%A8%D7%97%D7%91_%D7%95%D7%A7%D7%98%D7%95%D7%A8%D7%99" title="מרחב וקטורי">מרחב וקטורי</a> • <a href="/wiki/%D7%9E%D7%A9%D7%95%D7%95%D7%90%D7%94_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA" title="משוואה ליניארית">משוואה ליניארית</a> • <a href="/wiki/%D7%9E%D7%A2%D7%A8%D7%9B%D7%AA_%D7%9E%D7%A9%D7%95%D7%95%D7%90%D7%95%D7%AA_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%95%D7%AA" title="מערכת משוואות ליניאריות">מערכת משוואות ליניאריות</a> • <a class="mw-selflink selflink">העתקה ליניארית</a> • <a href="/wiki/%D7%9E%D7%98%D7%A8%D7%99%D7%A6%D7%94" title="מטריצה">מטריצה</a> </td></tr> <tr> <td style="background-color: #F2F3F4; text-align: right; font-weight: bold; padding-left: 5px;"><a href="/wiki/%D7%95%D7%A7%D7%98%D7%95%D7%A8_(%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94)" class="mw-redirect" title="וקטור (אלגברה)">וקטורים</a> </td> <td style="padding-right: 5px; text-align: right;"><a href="/wiki/%D7%A1%D7%A7%D7%9C%D7%A8_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="סקלר (מתמטיקה)">סקלר</a> • <a href="/wiki/%D7%9B%D7%A4%D7%9C_%D7%91%D7%A1%D7%A7%D7%9C%D7%A8" title="כפל בסקלר">כפל בסקלר</a> • <a href="/wiki/%D7%A6%D7%99%D7%A8%D7%95%D7%A3_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99" title="צירוף ליניארי">צירוף ליניארי</a> • <a href="/wiki/%D7%AA%D7%9C%D7%95%D7%AA_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA" title="תלות ליניארית">תלות ליניארית</a> • <a href="/wiki/%D7%A7%D7%91%D7%95%D7%A6%D7%94_%D7%A4%D7%95%D7%A8%D7%A9%D7%AA" title="קבוצה פורשת">קבוצה פורשת</a> • <a href="/wiki/%D7%91%D7%A1%D7%99%D7%A1_(%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94)" title="בסיס (אלגברה)">בסיס</a> • <a href="/wiki/%D7%A7%D7%95%D7%90%D7%95%D7%A8%D7%93%D7%99%D7%A0%D7%98%D7%95%D7%AA_(%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94)" title="קואורדינטות (אלגברה)">וקטור קואורדינטות</a> • <a href="/wiki/%D7%9E%D7%9E%D7%93_(%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA)" title="ממד (אלגברה ליניארית)">ממד</a> </td></tr> <tr> <td style="background-color: #F2F3F4; text-align: right; font-weight: bold; padding-left: 5px;"><a href="/wiki/%D7%9E%D7%98%D7%A8%D7%99%D7%A6%D7%94" title="מטריצה">מטריצות</a> </td> <td style="padding-right: 5px; text-align: right;"><a href="/wiki/%D7%9B%D7%A4%D7%9C_%D7%9E%D7%98%D7%A8%D7%99%D7%A6%D7%95%D7%AA" title="כפל מטריצות">כפל מטריצות</a> • <a href="/wiki/%D7%A9%D7%97%D7%9C%D7%95%D7%A3_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="שחלוף (מתמטיקה)">שחלוף</a> • <a href="/wiki/%D7%93%D7%98%D7%A8%D7%9E%D7%99%D7%A0%D7%A0%D7%98%D7%94" title="דטרמיננטה">דטרמיננטה</a> • <a href="/wiki/%D7%93%D7%99%D7%A8%D7%95%D7%92_%D7%9E%D7%98%D7%A8%D7%99%D7%A6%D7%95%D7%AA" title="דירוג מטריצות">דירוג מטריצות</a> • <a href="/wiki/%D7%93%D7%A8%D7%92%D7%94_(%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA)" title="דרגה (אלגברה ליניארית)">דרגה</a> • <a href="/wiki/%D7%A2%D7%A7%D7%91%D7%94_(%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94)" title="עקבה (אלגברה)">עקבה</a> • <a href="/wiki/%D7%9E%D7%98%D7%A8%D7%99%D7%A6%D7%94_%D7%9E%D7%A6%D7%95%D7%A8%D7%A4%D7%AA" title="מטריצה מצורפת">מטריצה מצורפת</a> • <a href="/wiki/%D7%9E%D7%98%D7%A8%D7%99%D7%A6%D7%AA_%D7%9E%D7%A2%D7%91%D7%A8" title="מטריצת מעבר">מטריצת מעבר</a> • <a href="/wiki/%D7%9E%D7%98%D7%A8%D7%99%D7%A6%D7%94_%D7%9E%D7%A9%D7%95%D7%9C%D7%A9%D7%99%D7%AA" title="מטריצה משולשית">מטריצה משולשית</a> • <a href="/wiki/%D7%93%D7%9E%D7%99%D7%95%D7%9F_%D7%9E%D7%98%D7%A8%D7%99%D7%A6%D7%95%D7%AA" title="דמיון מטריצות">דמיון מטריצות</a> • <a href="/wiki/%D7%A2%D7%A8%D7%9A_%D7%A2%D7%A6%D7%9E%D7%99" title="ערך עצמי">ערך עצמי</a> • <a href="/wiki/%D7%A4%D7%95%D7%9C%D7%99%D7%A0%D7%95%D7%9D_%D7%90%D7%95%D7%A4%D7%99%D7%99%D7%A0%D7%99" title="פולינום אופייני">פולינום אופייני</a> • <a href="/wiki/%D7%9E%D7%98%D7%A8%D7%99%D7%A6%D7%94_%D7%9C%D7%9B%D7%A1%D7%99%D7%A0%D7%94" title="מטריצה לכסינה">לכסון מטריצות</a> • <a href="/wiki/%D7%A6%D7%95%D7%A8%D7%AA_%D7%96%27%D7%95%D7%A8%D7%93%D7%9F" title="צורת ז'ורדן">צורת ז'ורדן</a> </td></tr> <tr> <td style="background-color: #F2F3F4; text-align: right; font-weight: bold; padding-left: 5px;"><a class="mw-selflink selflink">העתקות</a> </td> <td style="padding-right: 5px; text-align: right;"><a class="mw-selflink selflink">העתקה ליניארית</a> • <a href="/wiki/%D7%A7%D7%95%D7%90%D7%95%D7%A8%D7%93%D7%99%D7%A0%D7%98%D7%95%D7%AA_(%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94)" title="קואורדינטות (אלגברה)">קואורדינטות</a> • <a href="/wiki/%D7%9E%D7%98%D7%A8%D7%99%D7%A6%D7%94_%D7%9E%D7%99%D7%99%D7%A6%D7%92%D7%AA" class="mw-redirect" title="מטריצה מייצגת">מטריצה מייצגת</a> • <a href="/wiki/%D7%92%D7%A8%D7%A2%D7%99%D7%9F_(%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94)" title="גרעין (אלגברה)">גרעין</a> • <a href="/wiki/%D7%90%D7%A0%D7%93%D7%95%D7%9E%D7%95%D7%A8%D7%A4%D7%99%D7%96%D7%9D" title="אנדומורפיזם">אנדומורפיזם</a> • <a href="/wiki/%D7%90%D7%99%D7%96%D7%95%D7%9E%D7%95%D7%A8%D7%A4%D7%99%D7%96%D7%9D" title="איזומורפיזם">איזומורפיזם</a> • <a href="/wiki/%D7%94%D7%A2%D7%AA%D7%A7%D7%94_%D7%90%D7%A4%D7%99%D7%A0%D7%99%D7%AA" title="העתקה אפינית">העתקה אפינית</a> • <a href="/wiki/%D7%94%D7%A2%D7%AA%D7%A7%D7%94_%D7%A4%D7%A8%D7%95%D7%99%D7%A7%D7%98%D7%99%D7%91%D7%99%D7%AA" title="העתקה פרויקטיבית">העתקה פרויקטיבית</a> </td></tr> <tr> <td style="background-color: #F2F3F4; text-align: right; font-weight: bold; padding-left: 5px;"><a href="/wiki/%D7%9E%D7%A8%D7%97%D7%91_%D7%9E%D7%9B%D7%A4%D7%9C%D7%94_%D7%A4%D7%A0%D7%99%D7%9E%D7%99%D7%AA" title="מרחב מכפלה פנימית">מרחבי מכפלה פנימית</a> </td> <td style="padding-right: 5px; text-align: right;"><a href="/wiki/%D7%9E%D7%9B%D7%A4%D7%9C%D7%94_%D7%A1%D7%A7%D7%9C%D7%A8%D7%99%D7%AA" title="מכפלה סקלרית">מכפלה סקלרית</a> • <a href="/wiki/%D7%9E%D7%9B%D7%A4%D7%9C%D7%94_%D7%95%D7%A7%D7%98%D7%95%D7%A8%D7%99%D7%AA" title="מכפלה וקטורית">מכפלה וקטורית</a> • <a href="/wiki/%D7%90%D7%95%D7%A8%D7%AA%D7%95%D7%92%D7%95%D7%A0%D7%9C%D7%99%D7%95%D7%AA" title="אורתוגונליות">אורתוגונליות</a> • <a href="/wiki/%D7%9E%D7%98%D7%A8%D7%99%D7%A6%D7%94_%D7%A1%D7%99%D7%9E%D7%98%D7%A8%D7%99%D7%AA" title="מטריצה סימטרית">מטריצה סימטרית</a> • <a href="/wiki/%D7%90%D7%95%D7%A4%D7%A8%D7%98%D7%95%D7%A8_%D7%94%D7%A8%D7%9E%D7%99%D7%98%D7%99" title="אופרטור הרמיטי">אופרטור הרמיטי</a> • <a href="/wiki/%D7%90%D7%95%D7%A4%D7%A8%D7%98%D7%95%D7%A8_%D7%90%D7%95%D7%A0%D7%99%D7%98%D7%A8%D7%99" title="אופרטור אוניטרי">אופרטור אוניטרי</a> • <a href="/wiki/%D7%98%D7%A8%D7%A0%D7%A1%D7%A4%D7%95%D7%A8%D7%9E%D7%A6%D7%99%D7%94_%D7%A0%D7%95%D7%A8%D7%9E%D7%9C%D7%99%D7%AA" class="mw-redirect" title="טרנספורמציה נורמלית">טרנספורמציה נורמלית</a> • <a href="/wiki/%D7%A0%D7%95%D7%A8%D7%9E%D7%94_(%D7%90%D7%A0%D7%9C%D7%99%D7%96%D7%94)" title="נורמה (אנליזה)">נורמה</a> • <a href="/wiki/%D7%9E%D7%98%D7%A8%D7%99%D7%A7%D7%94" title="מטריקה">מטריקה</a> </td></tr> <tr> <td style="background-color: #F2F3F4; text-align: right; font-weight: bold; padding-left: 5px;"><a href="/wiki/%D7%AA%D7%91%D7%A0%D7%99%D7%AA_(%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA)" title="תבנית (אלגברה ליניארית)">תבניות</a> </td> <td style="padding-right: 5px; text-align: right;"><a href="/wiki/%D7%AA%D7%91%D7%A0%D7%99%D7%AA_%D7%91%D7%99%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA" title="תבנית ביליניארית">תבנית ביליניארית</a> • <a href="/wiki/%D7%AA%D7%91%D7%A0%D7%99%D7%AA_%D7%A1%D7%99%D7%9E%D7%98%D7%A8%D7%99%D7%AA" class="mw-redirect" title="תבנית סימטרית">תבנית סימטרית</a> • <a href="/w/index.php?title=%D7%AA%D7%91%D7%A0%D7%99%D7%AA_%D7%94%D7%A8%D7%9E%D7%99%D7%98%D7%99%D7%AA&action=edit&redlink=1" class="new" title="תבנית הרמיטית (הדף אינו קיים)">תבנית הרמיטית</a> • <a href="/w/index.php?title=%D7%AA%D7%91%D7%A0%D7%99%D7%AA_%D7%A1%D7%99%D7%9E%D7%A4%D7%9C%D7%A7%D7%98%D7%99%D7%AA_(%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA)&action=edit&redlink=1" class="new" title="תבנית סימפלקטית (אלגברה ליניארית) (הדף אינו קיים)">תבנית סימפלקטית</a> • <a href="/wiki/%D7%97%D7%A4%D7%99%D7%A4%D7%AA_%D7%9E%D7%98%D7%A8%D7%99%D7%A6%D7%95%D7%AA" title="חפיפת מטריצות">חפיפת מטריצות</a> • <a href="/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%94%D7%AA%D7%9E%D7%93%D7%94_%D7%A9%D7%9C_%D7%A1%D7%99%D7%9C%D7%91%D7%A1%D7%98%D7%A8" title="משפט ההתמדה של סילבסטר">משפט סילבסטר</a> • <a href="/w/index.php?title=%D7%AA%D7%91%D7%A0%D7%99%D7%AA_%D7%9E%D7%95%D7%9C%D7%98%D7%99-%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA_%D7%90%D7%A0%D7%98%D7%99-%D7%A1%D7%99%D7%9E%D7%98%D7%A8%D7%99%D7%AA&action=edit&redlink=1" class="new" title="תבנית מולטי-ליניארית אנטי-סימטרית (הדף אינו קיים)">תבנית מולטי-ליניארית אנטי-סימטרית</a> • <a href="/wiki/%D7%90%D7%95%D7%A8%D7%99%D7%99%D7%A0%D7%98%D7%A6%D7%99%D7%94_(%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA)" title="אוריינטציה (אלגברה ליניארית)">אוריינטציה</a> • <a href="/w/index.php?title=%D7%A6%D7%A4%D7%99%D7%A4%D7%95%D7%AA_(%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94_%D7%9C%D7%99%D7%A0%D7%99%D6%BC%D7%90%D7%A8%D7%99%D7%AA)&action=edit&redlink=1" class="new" title="צפיפות (אלגברה ליניּארית) (הדף אינו קיים)">צפיפות</a> • <a href="/wiki/%D7%98%D7%A0%D7%96%D7%95%D7%A8" title="טנזור">טנזור</a> </td></tr> </tbody></table> <div class="mw-heading mw-heading2"><h2 id="קישורים_חיצוניים"><span id=".D7.A7.D7.99.D7.A9.D7.95.D7.A8.D7.99.D7.9D_.D7.97.D7.99.D7.A6.D7.95.D7.A0.D7.99.D7.99.D7.9D"></span>קישורים חיצוניים</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%94%D7%A2%D7%AA%D7%A7%D7%94_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA&action=edit&section=11" title="עריכת קוד המקור של הפרק: קישורים חיצוניים"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%94%D7%A2%D7%AA%D7%A7%D7%94_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA&veaction=edit&section=11" title="עריכת פסקה: "קישורים חיצוניים"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="sisterwikilinkT"><span typeof="mw:File"><a href="https://commons.wikimedia.org/wiki/" title="ויקישיתוף"><img alt="ויקישיתוף" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/15px-Commons-logo.svg.png" decoding="async" width="15" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/23px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span> מדיה וקבצים בנושא <b><a href="https://commons.wikimedia.org/wiki/Category:Linear_operators" class="extiw" title="commons:Category:Linear operators">העתקה ליניארית</a></b> ב<a href="/wiki/%D7%95%D7%99%D7%A7%D7%99%D7%A9%D7%99%D7%AA%D7%95%D7%A3" title="ויקישיתוף">וויקישיתוף</a></div> <ul><li><a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/LinearTransformation.html">העתקה ליניארית</a>, באתר <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a> <span dir="rtl" class="languageicon">(באנגלית)</span><style data-mw-deduplicate="TemplateStyles:r36549940">.mw-parser-output .languageicon{font-size:0.95em;font-weight:bold;color:#555}</style></li> <li><span typeof="mw:File"><a href="/wiki/%D7%A7%D7%95%D7%91%D7%A5:National_Library_IL_logo.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/National_Library_IL_logo.svg/20px-National_Library_IL_logo.svg.png" decoding="async" width="20" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/National_Library_IL_logo.svg/30px-National_Library_IL_logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/df/National_Library_IL_logo.svg/40px-National_Library_IL_logo.svg.png 2x" data-file-width="512" data-file-height="397" /></a></span> <a rel="nofollow" class="external text" href="https://www.nli.org.il/he/a-topic/987007529341405171">אופרטורים ליניאריים</a>, דף שער ב<a href="/wiki/%D7%94%D7%A1%D7%A4%D7%A8%D7%99%D7%99%D7%94_%D7%94%D7%9C%D7%90%D7%95%D7%9E%D7%99%D7%AA" title="הספרייה הלאומית">ספרייה הלאומית</a></li></ul> <div role="navigation" class="navbox authority-control" aria-labelledby="בקרת_זהויות_15px&#124;link=https&#58;//www.wikidata.org/wiki/Q207643?uselang=he&#124;עריכת_הנתון_בוויקינתונים" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th id="בקרת_זהויות_15px&#124;link=https&#58;//www.wikidata.org/wiki/Q207643?uselang=he&#124;עריכת_הנתון_בוויקינתונים" scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%D7%A2%D7%96%D7%A8%D7%94:%D7%91%D7%A7%D7%A8%D7%AA_%D7%96%D7%94%D7%95%D7%99%D7%95%D7%AA" title="עזרה:בקרת זהויות">בקרת זהויות</a> <span typeof="mw:File"><a href="https://www.wikidata.org/wiki/Q207643?uselang=he" title="עריכת הנתון בוויקינתונים"><img alt="עריכת הנתון בוויקינתונים" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Blue_pencil_RTL.svg/15px-Blue_pencil_RTL.svg.png" decoding="async" width="15" height="15" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Blue_pencil_RTL.svg/23px-Blue_pencil_RTL.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Blue_pencil_RTL.svg/30px-Blue_pencil_RTL.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></th><td class="navbox-list navbox-odd" style="text-align:left;border-right-width:2px;border-right-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><div dir="ltr"> <ul><li><span class="nowrap"><a href="/wiki/%D7%94%D7%A1%D7%A4%D7%A8%D7%99%D7%99%D7%94_%D7%94%D7%9C%D7%90%D7%95%D7%9E%D7%99%D7%AA" title="הספרייה הלאומית">NLI</a>: <span class="uid"><a rel="nofollow" class="external text" href="http://olduli.nli.org.il/F/?func=find-b&local_base=NLX10&find_code=UID&request=987007529341405171">987007529341405171</a></span></span></li> <li><span class="nowrap"><a href="/wiki/%D7%94%D7%A1%D7%A4%D7%A8%D7%99%D7%99%D7%94_%D7%94%D7%9C%D7%90%D7%95%D7%9E%D7%99%D7%AA_%D7%A9%D7%9C_%D7%A6%D7%A8%D7%A4%D7%AA" title="הספרייה הלאומית של צרפת">BnF</a>: <span class="uid"><a rel="nofollow" class="external text" href="https://catalogue.bnf.fr/ark:/12148/cb11944511p">cb11944511p</a> <a rel="nofollow" class="external text" href="https://data.bnf.fr/ark:/12148/cb11944511p">(data)</a></span></span></li> <li><span class="nowrap"><a href="/wiki/%D7%A7%D7%95%D7%91%D7%A5_%D7%91%D7%A7%D7%A8%D7%94_%D7%9E%D7%A9%D7%95%D7%9C%D7%91" title="קובץ בקרה משולב">GND</a>: <span class="uid"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4167700-6">4167700-6</a></span></span></li> <li><span class="nowrap"><a href="/wiki/%D7%A9%D7%99%D7%98%D7%AA_%D7%A1%D7%A4%D7%A8%D7%99%D7%99%D7%AA_%D7%94%D7%A7%D7%95%D7%A0%D7%92%D7%A8%D7%A1" title="שיטת ספריית הקונגרס">LCCN</a>: <span class="uid"><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/subjects/sh85077178">sh85077178</a></span></span></li></ul> </div></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">אוחזר מתוך "<a dir="ltr" href="https://he.wikipedia.org/w/index.php?title=העתקה_ליניארית&oldid=39819498">https://he.wikipedia.org/w/index.php?title=העתקה_ליניארית&oldid=39819498</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/%D7%95%D7%99%D7%A7%D7%99%D7%A4%D7%93%D7%99%D7%94:%D7%A7%D7%98%D7%92%D7%95%D7%A8%D7%99%D7%94" title="ויקיפדיה:קטגוריה">קטגוריות</a>: <ul><li><a href="/wiki/%D7%A7%D7%98%D7%92%D7%95%D7%A8%D7%99%D7%94:%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA" title="קטגוריה:אלגברה 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title="קטגוריה:ויקיפדיה: ערכים עם מזהה J9U">ויקיפדיה: ערכים עם מזהה J9U</a></li><li><a href="/wiki/%D7%A7%D7%98%D7%92%D7%95%D7%A8%D7%99%D7%94:%D7%95%D7%99%D7%A7%D7%99%D7%A4%D7%93%D7%99%D7%94:_%D7%A2%D7%A8%D7%9B%D7%99%D7%9D_%D7%A2%D7%9D_%D7%9E%D7%96%D7%94%D7%94_BNF" title="קטגוריה:ויקיפדיה: ערכים עם מזהה BNF">ויקיפדיה: ערכים עם מזהה BNF</a></li><li><a href="/wiki/%D7%A7%D7%98%D7%92%D7%95%D7%A8%D7%99%D7%94:%D7%95%D7%99%D7%A7%D7%99%D7%A4%D7%93%D7%99%D7%94:_%D7%A2%D7%A8%D7%9B%D7%99%D7%9D_%D7%A2%D7%9D_%D7%9E%D7%96%D7%94%D7%94_GND" title="קטגוריה:ויקיפדיה: ערכים עם מזהה GND">ויקיפדיה: ערכים עם מזהה GND</a></li><li><a href="/wiki/%D7%A7%D7%98%D7%92%D7%95%D7%A8%D7%99%D7%94:%D7%95%D7%99%D7%A7%D7%99%D7%A4%D7%93%D7%99%D7%94:_%D7%A2%D7%A8%D7%9B%D7%99%D7%9D_%D7%A2%D7%9D_%D7%9E%D7%96%D7%94%D7%94_LCCN" title="קטגוריה:ויקיפדיה: ערכים עם מזהה LCCN">ויקיפדיה: ערכים עם מזהה LCCN</a></li></ul></div></div> </div> </main> <div id='mw-data-after-content'> <div class="read-more-container"></div> </div> </div> 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