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.
Best Answer
Just write it down. That is, given a basis $\{ e_i \ | i = 1..n \}$ of $V$ and $\{ f_i \ | i = 1..n \}$ of $W$ define $T : V \rightarrow W$ first on the basis vectors,
$$T(e_i) = f_i \ \ \ \ \text{ for each } i = 1, 2, ..., n$$
Now how would you extend $T$ to all of $V$?