A basic question on isomorphism of two vector spaces Let $V$ and $W$ be of same dimension (over the same field), say dim$V$=dim$W$ = $k < \infty$. Let $T : V \rightarrow W$ be a linear map (which respects vector addition and scalar multiplication). Then if $T$ is $1-1$ then $T$ is onto. How to prove it ? Also, how to prove other way ? i.e. $T$ is onto implies $T$ is $1-1$
Regarding the first part, any $w \in W$ can be written as linear combination of basis vectors of $W$ say $w_i, i=1,\dots,k$, but I can't go to the set $V$ using this $w_i$ because I don't know whether $T$ is onto or not. How to proceed from here ?

This result relies on the following fact. If $T:V\to W$ is a linear transformation between any two vector spaces, then the dimension $\dim(V)=\dim(\ker(T))+\dim(\operatorname{im}(T))$.
You'll also need to know that $T$ is injective iff its kernel is $\{0\}$, and that a subspace $W'$ of $W$ of the same dimension as $W$ is $W$. 
