Basis Selection Lemma

linear-algebra

Definition

Basis Selection Lemma

Let $M=\{{\mathbf{v}}_1,\dots ,{\mathbf{v}}_n\}$ be the basis of a vector space.

$$\mathbf{a}={\mu }_1\cdot {\mathbf{v}}_1+\cdots +{\mu }_n\cdot {\mathbf{v}}_n$$

a linear combination of $M$.

Further, let $M^{\prime }$ be:

$$\begin{array}{rl}{M}^{\prime }& =(M\setminus \{{\mathbf{v}}_j\})\cup \{\mathbf{a}\}\\ & =\{{\mathbf{v}}_1,\dots ,{\mathbf{v}}_{j-1},\mathbf{a},{\mathbf{v}}_{j+1},\dots ,{\mathbf{v}}_n\}\end{array}$$

for $j$ with ${\mu }_j\ne 0$.

Then $M$ is linearly independent if and only if $M^{\prime }$ is linearly independent, and it always holds that:

$$[M^{\prime }]=[M]$$

where $[\cdot ]$ denotes the linear span.