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## Homework Statement

A theorem in my book states: If V, W are finite dimensional vector spaces that are isomorphic, then V, W have the same dimension. I wrote a proof but it is different from the proof given in my book, and I'd like to know if it's right.

## The Attempt at a Solution

Let [itex]\left\{A_1, ...,A_n\right\}[/itex] be a basis for V. Because V,W are isomorphic, we know [itex]ImT = W[/itex], which implies that [itex]dim(ImT) = dim(W)[/itex]. We also know [itex]KerT = \left\{ 0 \right\}[/itex] and hence [itex]dim(KerT) = 0[/itex] We know [itex]dim(V) = n = dim(ImT) + dim(KerT)= dim(ImT) + 0 = dim(ImT)[/itex]. Thus, [itex]dim(ImT) = n = dim(W)[/itex]

Is that right?