Joyce Smith

2022-01-04

Let V and W be vector spaces, let $T:V\to W$ be linear, and let $\{{w}_{1},{w}_{2},\dots ,{w}_{k}\}$ be a linearly independent set of k vectors from R(T). Prove that if $S=\{{v}_{1},{v}_{2},...,{v}_{k}\}$ is chosen so that $T\left({v}_{i}\right)={W}_{i}$ for $i=1,2,\dots ,k,$ then S is linearly independent.

Daniel Cormack

Beginner2022-01-05Added 34 answers

Definitions:

1) The set of vectors

The map

(i)

(ii)

Let us consider an arbitrary representation of zero vector of V as a linear combination of vectors from S.

Since T is linear,

i.e.,

Thus,

By the linearity of T we have

since the set

substituting

Thus (1) will be the trivial representation of zero vector of V.

Hence

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