# 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 |

**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