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_{1},....v

_{k}} be a subset of V. Prove that {v

_{1},....v

_{k}} is a linearly independent set iff {T(v

_{1}),....T(v

_{k})} is a linearly independent set.

Attempt: All I know is that {T(v

_{1}),....T(v

_{k})} means that each set of co-ordinate vectors can be written as a linear combination of the standard basis vectors, which means the co-ordinate vectors are linearly independent. As well if we are assuming T is isomorphic then T has an inverse. How or what other facts am I suppose to use?

I'm at one of those boiling points of frustration when it comes to these proofs. As if I really don't know how to construct them. What am I missing when it comes to constructing these things. Sincerely, ABout to blow a gasket.