Vector spaces of the same finite dimension are isomorphic 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. 
 A: When you have two vector spaces $V$ and $W$, if you are given a basis $\{v_1,v_2,\dots,v_n\}$ of $V$ and any set $\{w_1,w_2,\dots,w_n\}$, there is a unique linear map $f\colon V\to W$ such that
$$
f(v_i)=w_i\quad (i=1,2,\dots,n)
$$
How's it defined? Since every vector $v\in V$ can be written in one and only one way  as
$$
v=\alpha_1v_1+\dots+\alpha_nv_n,
$$
we define
$$
f(v)=\alpha_1w_1+\dots+\alpha_nw_n.
$$
The verification that this is a linear map are just tedious, but basically easy.
If moreover $\{w_1,w_2,\dots,w_n\}$ is a basis of $W$, the map $f$ so defined maps a basis to a basis, so it's an isomorphism.
A: Denote the dimension by $n$ and let $(v_1,\ldots,v_n)$ a basis for $V$ and $(w_1,\ldots,w_n)$ a basis for $W$ and let the linear transformation $f:V\to W$ defined by
$$f(v_i)=w_i$$
then $f$ transforms a basis to a basis then $f$ is an isomorphism of vector spaces.
A: List your bases $\{v_1, \ldots , v_n \}$ for $V$ and $\{w_1, \ldots, w_n\}$ for $W$. I'm assuming these are both vector spaces over the same field $F$. Suppose that $\sum_{i = 1}^n a_i v_i \in V$ for some $a_i \in F$. Then define $\phi : V \to W$ by $$ \phi \big(\sum_{i = 1}^n a_i v_i\big) = \sum_{i = 1}^n a_i w_i $$
Not too hard to see it's a well defined  vector space isomorphism.
A: 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 $V$?
