Suppose that V and W are vector spaces with the same dimension. We wish to show that V is isomorphic to W, i.e. show that there exists a bijective linear function, mapping from V to W.
I understand that it will suffice to find a linear function that maps a basis of V to a basis of W. This is because any element of a vector space can be written as a unique linear combination of its basis elements.
However I'm not sure how to show that such a map exists.