Kernel of a matrix

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