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.mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is about zeros of a quadratic form. For the zero element in a vector space, see <a href="/wiki/Zero_vector" class="mw-redirect" title="Zero vector">Zero vector</a>. For null vectors in Minkowski space, see <a href="/wiki/Minkowski_space#Causal_structure" title="Minkowski space">Minkowski space § Causal structure</a>.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Conformalsphere.pdf" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Conformalsphere.pdf/page1-220px-Conformalsphere.pdf.jpg" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Conformalsphere.pdf/page1-330px-Conformalsphere.pdf.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Conformalsphere.pdf/page1-440px-Conformalsphere.pdf.jpg 2x" data-file-width="1156" data-file-height="1156" /></a><figcaption>A null cone where <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 q(x,y,z)=x^{2}+y^{2}-z^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q(x,y,z)=x^{2}+y^{2}-z^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fef1cf16b809ee7ab14a3e6840f3e69de6444c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.689ex; height:3.176ex;" alt="{\displaystyle q(x,y,z)=x^{2}+y^{2}-z^{2}.}"></span></figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, given a <a href="/wiki/Vector_space" title="Vector space">vector space</a> <i>X</i> with an associated <a href="/wiki/Quadratic_form" title="Quadratic form">quadratic form</a> <i>q</i>, written <span class="nowrap">(<i>X</i>, <i>q</i>)</span>, a <b>null vector</b> or <b>isotropic vector</b> is a non-zero element <i>x</i> of <i>X</i> for which <span class="nowrap"><i>q</i>(<i>x</i>) = 0</span>. </p><p>In the theory of <a href="/wiki/Real_number" title="Real number">real</a> <a href="/wiki/Bilinear_form" title="Bilinear form">bilinear forms</a>, <a href="/wiki/Definite_quadratic_form" title="Definite quadratic form">definite quadratic forms</a> and <a href="/wiki/Isotropic_quadratic_form" title="Isotropic quadratic form">isotropic quadratic forms</a> are distinct. They are distinguished in that only for the latter does there exist a nonzero null vector. </p><p>A <a href="/wiki/Quadratic_space" class="mw-redirect" title="Quadratic space">quadratic space</a> <span class="nowrap">(<i>X</i>, <i>q</i>)</span> which has a null vector is called a <a href="/wiki/Pseudo-Euclidean_space" title="Pseudo-Euclidean space">pseudo-Euclidean space</a>. The term <i>isotropic vector v</i> when <i>q</i>(<i>v</i>) = 0 has been used in quadratic spaces,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> and <b>anisotropic space</b> for a quadratic space without null vectors. </p><p>A pseudo-Euclidean vector space may be decomposed (non-uniquely) into <a href="/wiki/Orthogonal_subspaces" class="mw-redirect" title="Orthogonal subspaces">orthogonal subspaces</a> <i>A</i> and <i>B</i>, <span class="nowrap"><i>X</i> = <i>A</i> + <i>B</i></span>, where <i>q</i> is positive-definite on <i>A</i> and negative-definite on <i>B</i>. The <b>null cone</b>, or <b>isotropic cone</b>, of <i>X</i> consists of the union of balanced spheres: <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 \bigcup _{r\geq 0}\{x=a+b:q(a)=-q(b)=r,\ \ a,b\in B\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mo>≥</mo> <mn>0</mn> </mrow> </munder> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>:</mo> <mi>q</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>q</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>r</mi> <mo>,</mo> <mtext> </mtext> <mtext> </mtext> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mi>B</mi> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigcup _{r\geq 0}\{x=a+b:q(a)=-q(b)=r,\ \ a,b\in B\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3639e6b4596081f0c5af454dba11a4a2dda376ea" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:44.347ex; height:5.676ex;" alt="{\displaystyle \bigcup _{r\geq 0}\{x=a+b:q(a)=-q(b)=r,\ \ a,b\in B\}.}"></span> The null cone is also the union of the <a href="/wiki/Isotropic_line" title="Isotropic line">isotropic lines</a> through the origin. </p> <div class="mw-heading mw-heading2"><h2 id="Split_algebras">Split algebras</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Null_vector&action=edit&section=1" title="Edit section: Split algebras"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A composition algebra with a null vector is a <b>split algebra</b>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>In a <a href="/wiki/Composition_algebra" title="Composition algebra">composition algebra</a> (<i>A</i>, +, ×, *), the quadratic form is q(<i>x</i>) = <i>x x</i>*. When <i>x</i> is a null vector then there is no multiplicative inverse for <i>x</i>, and since <i>x</i> ≠ 0, <i>A</i> is not a <a href="/wiki/Division_algebra" title="Division algebra">division algebra</a>. </p><p>In the <a href="/wiki/Cayley%E2%80%93Dickson_construction" title="Cayley–Dickson construction">Cayley–Dickson construction</a>, the split algebras arise in the series <a href="/wiki/Bicomplex_number" title="Bicomplex number">bicomplex numbers</a>, <a href="/wiki/Biquaternion" title="Biquaternion">biquaternions</a>, and <a href="/wiki/Bioctonion" title="Bioctonion">bioctonions</a>, which uses the <a href="/wiki/Complex_number" title="Complex number">complex number</a> field <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 {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span> as the foundation of this doubling construction due to <a href="/wiki/L._E._Dickson" class="mw-redirect" title="L. E. Dickson">L. E. Dickson</a> (1919). In particular, these algebras have two <a href="/wiki/Imaginary_unit" title="Imaginary unit">imaginary units</a>, which commute so their product, when squared, yields +1: </p> <dl><dd><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 (hi)^{2}=h^{2}i^{2}=(-1)(-1)=+1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>h</mi> <mi>i</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo>+</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (hi)^{2}=h^{2}i^{2}=(-1)(-1)=+1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a972dae66835f1ddbfb0709775f2598527695a48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.727ex; height:3.176ex;" alt="{\displaystyle (hi)^{2}=h^{2}i^{2}=(-1)(-1)=+1.}"></span> Then</dd> <dd><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 (1+hi)(1+hi)^{*}=(1+hi)(1-hi)=1-(hi)^{2}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>h</mi> <mi>i</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>h</mi> <mi>i</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>h</mi> <mi>i</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>h</mi> <mi>i</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mo stretchy="false">(</mo> <mi>h</mi> <mi>i</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1+hi)(1+hi)^{*}=(1+hi)(1-hi)=1-(hi)^{2}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a56f59d11c9204244cae2a762297111450041603" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:52.334ex; height:3.176ex;" alt="{\displaystyle (1+hi)(1+hi)^{*}=(1+hi)(1-hi)=1-(hi)^{2}=0}"></span> so 1 + hi is a null vector.</dd></dl> <p>The real subalgebras, <a href="/wiki/Split_complex_number" class="mw-redirect" title="Split complex number">split complex numbers</a>, <a href="/wiki/Split_quaternion" class="mw-redirect" title="Split quaternion">split quaternions</a>, and <a href="/wiki/Split-octonion" title="Split-octonion">split-octonions</a>, with their null cones representing the light tracking into and out of 0 ∈ <i>A</i>, suggest <a href="/wiki/Spacetime_topology" title="Spacetime topology">spacetime topology</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Null_vector&action=edit&section=2" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Minkowski_space#Causal_structure" title="Minkowski space">light-like</a> vectors of <a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski space</a> are null vectors. </p><p>The four <a href="/wiki/Linearly_independent" class="mw-redirect" title="Linearly independent">linearly independent</a> <a href="/wiki/Biquaternion" title="Biquaternion">biquaternions</a> <span class="nowrap"><i>l</i> = 1 + <i>hi</i></span>, <span class="nowrap"><i>n</i> = 1 + <i>hj</i></span>, <span class="nowrap"><i>m</i> = 1 + <i>hk</i></span>, and <span class="nowrap"><i>m</i><sup>∗</sup> = 1 – <i>hk</i></span> are null vectors and <span class="nowrap">{ <i>l</i>, <i>n</i>, <i>m</i>, <i>m</i><sup>∗</sup> }</span> can serve as a <a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a> for the subspace used to represent <a href="/wiki/Spacetime" title="Spacetime">spacetime</a>. Null vectors are also used in the <a href="/wiki/Newman%E2%80%93Penrose_formalism" title="Newman–Penrose formalism">Newman–Penrose formalism</a> approach to spacetime manifolds.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>In the <a href="/wiki/Verma_module" title="Verma module">Verma module</a> of a <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebra</a> there are null vectors. </p> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Null_vector&action=edit&section=3" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a href="/wiki/Emil_Artin" title="Emil Artin">Emil Artin</a> (1957) <a href="/wiki/Geometric_Algebra_(book)" title="Geometric Algebra (book)"><i>Geometric Algebra</i></a>, <a rel="nofollow" class="external text" href="https://archive.org/details/geometricalgebra033556mbp/page/n129/mode/2up?view=theater&q=isotropic">isotropic</a></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Arthur A. Sagle & Ralph E. Walde (1973) <i>Introduction to Lie Groups and Lie Algebras</i>, page 197, <a href="/wiki/Academic_Press" title="Academic Press">Academic Press</a></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Patrick Dolan (1968) <a rel="nofollow" class="external text" href="http://projecteuclid.org/euclid.cmp/1103840725">A Singularity-free solution of the Maxwell-Einstein Equations</a>, <a href="/wiki/Communications_in_Mathematical_Physics" title="Communications in Mathematical Physics">Communications in Mathematical Physics</a> 9(2):161–8, especially 166, link from <a href="/wiki/Project_Euclid" title="Project Euclid">Project Euclid</a></span> </li> </ol></div></div> <ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFDubrovinFomenkoNovikov1984" class="citation book cs1">Dubrovin, B. A.; <a href="/wiki/Anatoly_Fomenko" title="Anatoly Fomenko">Fomenko, A. T.</a>; <a href="/wiki/Sergei_Novikov_(mathematician)" title="Sergei Novikov (mathematician)">Novikov, S. P.</a> (1984). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/moderngeometryme000dubr"><i>Modern Geometry: Methods and Applications</i></a></span>. Translated by Burns, Robert G. Springer. p. <a rel="nofollow" class="external text" href="https://archive.org/details/moderngeometryme000dubr/page/50">50</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-90872-2" title="Special:BookSources/0-387-90872-2"><bdi>0-387-90872-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Modern+Geometry%3A+Methods+and+Applications&rft.pages=50&rft.pub=Springer&rft.date=1984&rft.isbn=0-387-90872-2&rft.aulast=Dubrovin&rft.aufirst=B.+A.&rft.au=Fomenko%2C+A.+T.&rft.au=Novikov%2C+S.+P.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmoderngeometryme000dubr&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANull+vector" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShaw1982" class="citation book cs1">Shaw, Ronald (1982). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=C6DgAAAAMAAJ"><i>Linear Algebra and Group Representations</i></a>. Vol. 1. <a href="/wiki/Academic_Press" title="Academic Press">Academic Press</a>. p. 151. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-12-639201-3" title="Special:BookSources/0-12-639201-3"><bdi>0-12-639201-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra+and+Group+Representations&rft.pages=151&rft.pub=Academic+Press&rft.date=1982&rft.isbn=0-12-639201-3&rft.aulast=Shaw&rft.aufirst=Ronald&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DC6DgAAAAMAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANull+vector" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNeville1922" class="citation book cs1"><a href="/wiki/Eric_Harold_Neville" title="Eric Harold Neville">Neville, E. H. (Eric Harold)</a> (1922). <a rel="nofollow" class="external text" href="https://archive.org/details/prolegomenatoana00nevi"><i>Prolegomena to Analytical Geometry in Anisotropic Euclidean Space of Three Dimensions</i></a>. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. p. <a rel="nofollow" class="external text" href="https://archive.org/details/prolegomenatoana00nevi/page/204">204</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Prolegomena+to+Analytical+Geometry+in+Anisotropic+Euclidean+Space+of+Three+Dimensions&rft.pages=204&rft.pub=Cambridge+University+Press&rft.date=1922&rft.aulast=Neville&rft.aufirst=E.+H.+%28Eric+Harold%29&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fprolegomenatoana00nevi&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANull+vector" class="Z3988"></span></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Null_vector&oldid=1247890716">https://en.wikipedia.org/w/index.php?title=Null_vector&oldid=1247890716</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Linear_algebra" title="Category:Linear algebra">Linear algebra</a></li><li><a href="/wiki/Category:Quadratic_forms" title="Category:Quadratic forms">Quadratic forms</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 26 September 2024, at 15:33<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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