Basis Selection Lemma

linear-algebra

Definition

Basis Selection Lemma

Let $B=\{v_1,\dots ,v_n\}$ be a basis of a vector space $V$ and let

$$a=\sum _{i=1}^n{\mu }_i{v}_i$$

be a linear combination with ${\mu }_j\ne 0$ for some $j$. Then the exchanged set

$$B^{\prime }=(B\setminus \{v_j\})\cup \{a\}$$

is also a basis of $V$. In particular, $B^{\prime }$ is linearly independent and $[B^{\prime }]=[B]$, where $[\cdot ]$ denotes the linear span.