Isomorphisms preserve linear independence

[GlassBones]

Homework Statement

Let $T:V \rightarrow W$ be an ismorphism. Let $\{v_1, ..., v_k\}$ be a subset of V. Prove that $\{v_1, ..., v_k\}$ is a linearly independent set if and only if $\{T(v_1), ... , T(v_2)\}$ is a linearly independent set.

Homework Equations

The Attempt at a Solution

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$\rightarrow$: I began with the definition of linear independent vectors.
But I realized this could map to vectors that become dependent vectors in $W$.

I suppose the fact that T is an isomorphism is a hint. Can anyone give me ideas?

 

[Orodruin]

I suppose the fact that T is an isomorphism is a hint.

It is not a hint, it is a requirement and part of the question. In order to show that A holds iff B is true, then clearly the properties of B must somehow come into play.

So what are the properties of isomorphisms between vector spaces?

 

[fresh_42]

I suppose the fact that T is an isomorphism is a hint. Can anyone give me ideas?

A hint would be that injectivity is sufficient.