# Isomorphisms preserve linear independence

#### GlassBones

1. The problem statement, all variables and given/known data

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.

2. Relevant equations

3. The attempt at a solution

$\rightarrow$: 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?

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#### Orodruin

Staff Emeritus
Homework Helper
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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

Mentor
2018 Award
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.

"Isomorphisms preserve linear independence"

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