Linear independence

Overview

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Basic definition

• terminology to deal with a highly redundant

Definition: Linear independence

Let $F$ be a field and $V$ be an $F$-vector space. A sequence of vectors $w_1,\dots ,w_{\ell }\in V$ is linearly independent if whenever $t_1{w}_1+\cdots +t_{\ell }{w}_{\ell }=0$ for some $t_i\in F$, then $t_i=0$ for all $i$. We say a sequence is linearly dependent if this is not the case.

Note that a sequence $w_1,\dots ,w_{\ell }\in V$ with repeated vectors $w_i=w_j$ is always linearly dependent since we can express $0$ as the nontrivial linear combination $w_i-w_j$. Further, the sequences is automatically linearly dependent if $w_i=0$ for some $i$.

• Also not linearly dependent if we can write one vector as a linear combination of the others

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Code snippets

w_1, \ldots, w_\ell

t_1w_1 + \cdots + t_\ell w_\ell = 0