Exercise 21

Suppose that $f\colon V\to W$ is an isomorphism. Prove that the set $\text{$\{\hskip 1.00006pt\vec{v}_{1},\dots,\vec{v}_{k}\hskip 1.00006pt\}$}\subseteq V$ is linearly dependent if and only if the set of images $\text{$\{\hskip 1.00006ptf(\vec{v}_{1}),\dots,f(\vec{v}_{k})\hskip 1.00006pt\}$}\subseteq W$ is linearly dependent.