## Vladimir Dobrushkin

http://math.uri.edu/~dobrush/

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the appendix entitled GNU Free Documentation License.

# Kernels or Null Spaces

A **basis** β for a vector space *V* is a linearly independent subset of *V*
that generates

or span *V*. If β is a basis for *V*, we also say that elements of β form a basis for
*V*.

This means that every vector from *V* is a finite linear combination of elements from the basis.

Recall that a set of vectors β is said to generate or span a vector space *V* if every element from
*V* can be represented as a linear combination of vectors from β.

*n*-dimensional real space, and it is called the

**standard basis**. Its dimension is

*n*.

*i*-th row and

*j*-th column. Then the set \( {\bf M}_{i,j} \ : \ 1 \le i \le m , \ 1 \le j \le n \) is a basis for the set of all such real matrices. Its dimension is

*mn*.

*n*. It has dimension

*n*+1. ■

Theorem: Let *V* be a vector space and
\( \beta = \left\{ {\bf u}_1 , {\bf u}_2 , \ldots , {\bf u}_n \right\} \) be a subset of
*V*. Then β is a basis for *V* if and only if each vector *v* in *V* can be uniquely
decomposed into a linear combination of vectors in β, that is, can be uniquely expressed in the form