# Kernel of a matrix

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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=47489
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article