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A [[Matrix|matrix]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110090/k1100901.png" /> of size <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110090/k1100902.png" /> over a [[Field|field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110090/k1100903.png" /> defines a linear function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110090/k1100904.png" /> between the standard vector spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110090/k1100905.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110090/k1100906.png" /> by the well-known formula
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110090/k1100907.png" /></td> </tr></table>
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The kernel of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110090/k1100908.png" /> is the kernel of the linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110090/k1100909.png" />. The kernel of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110090/k11009010.png" /> (respectively, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110090/k11009011.png" />) is also called the null space or nullspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110090/k11009012.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110090/k11009013.png" />).
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A [[Matrix|matrix]]  $  A = ( a _ {ij }  ) $
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of size  $  n \times m $
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over a [[Field|field]]  $  K $
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defines a linear function  $  \alpha : {K  ^ {m} } \rightarrow {K  ^ {n} } $
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between the standard vector spaces  $  K  ^ {m} $
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and  $  K  ^ {n} $
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by the well-known formula
 +
 
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$$
 +
\alpha \left (
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 +
\begin{array}{c}
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v _ {1}  \\
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\vdots  \\
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v _ {m}  \\
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\end{array}
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\right ) = \left (
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\begin{array}{c}
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\sum a _ {1i }  v _ {i}  \\
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\vdots  \\
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\sum a _ {ni }  v _ {i}  \\
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\end{array}
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\right ) .
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$$
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The kernel of the matrix $  A $
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is the kernel of the linear mapping $  \alpha $.  
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The kernel of $  A $(
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respectively, of $  \alpha $)  
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is also called the null space or nullspace of $  A $(
 +
respectively, $  \alpha $).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Strang,  "Linear algebra and its applications" , Harcourt–Brace–Jovanovich  (1988)  pp. 92</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Schneider,  G.P. Barker,  "Matrices and linear algebra" , Dover, reprint  (1989)  pp. 215</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  B. Noble,  J.W. Daniel,  "Applied linear algebra" , Prentice-Hall  (1977)  pp. 157</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  Ch.G. Cullen,  "Matrices and linear transformations" , Dover, reprint  (1990)  pp. 187</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Strang,  "Linear algebra and its applications" , Harcourt–Brace–Jovanovich  (1988)  pp. 92</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Schneider,  G.P. Barker,  "Matrices and linear algebra" , Dover, reprint  (1989)  pp. 215</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  B. Noble,  J.W. Daniel,  "Applied linear algebra" , Prentice-Hall  (1977)  pp. 157</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  Ch.G. Cullen,  "Matrices and linear transformations" , Dover, reprint  (1990)  pp. 187</TD></TR></table>

Latest revision as of 22:14, 5 June 2020


A matrix $ A = ( a _ {ij } ) $ of size $ n \times m $ over a field $ K $ defines a linear function $ \alpha : {K ^ {m} } \rightarrow {K ^ {n} } $ between the standard vector spaces $ K ^ {m} $ and $ K ^ {n} $ by the well-known formula

$$ \alpha \left ( \begin{array}{c} v _ {1} \\ \vdots \\ v _ {m} \\ \end{array} \right ) = \left ( \begin{array}{c} \sum a _ {1i } v _ {i} \\ \vdots \\ \sum a _ {ni } v _ {i} \\ \end{array} \right ) . $$

The kernel of the matrix $ A $ is the kernel of the linear mapping $ \alpha $. The kernel of $ A $( respectively, of $ \alpha $) is also called the null space or nullspace of $ A $( respectively, $ \alpha $).

References

[a1] G. Strang, "Linear algebra and its applications" , Harcourt–Brace–Jovanovich (1988) pp. 92
[a2] H. Schneider, G.P. Barker, "Matrices and linear algebra" , Dover, reprint (1989) pp. 215
[a3] B. Noble, J.W. Daniel, "Applied linear algebra" , Prentice-Hall (1977) pp. 157
[a4] Ch.G. Cullen, "Matrices and linear transformations" , Dover, reprint (1990) pp. 187
How to Cite This Entry:
Kernel of a matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kernel_of_a_matrix&oldid=12040
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article