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Isomorphisms preserve linear independence

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?
 
Last edited:

Orodruin

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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

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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.
 

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