is linear independence preserved under isomorphism?

Question

Suppose $\phi:V \to W$ is an isomorphism. Let $\{v_k\}_{k=1}^n \subset V$ be linearly independent. Are $\{ \phi(v_k) \}_{k=1}^n \subset W$ linearly independent?

Answer 1

Note that, as $\phi$ is a morphism, for all $\lambda_1,\dots,\lambda_s\in\mathbb{K},$ $$\lambda_1\phi(v_1)+\dots+\lambda_s\phi(v_s)=\phi(\lambda_1v_1+\dots+\lambda_sv_s),$$ so $\lambda_1\phi(v_1)+\dots+\lambda_s\phi(v_s)=0$ implies $\lambda_1v_1+\dots+\lambda_sv_s=0$ (because $\phi$ is an isomorphism, and so injective as egreg said) which implies $\lambda_1=\dots=\lambda_s=0$ (because $v_1,\dots,v_s$ are linearly independant).