Problem about the image and kernel of a linear transformation Can someone help me with this problem on Linear Algebra.
Let $F$ be a field, $V$ and $W$ vector spaces over $F$. Prove that, if there exist a linear transformation $T:V\to W$ whose image and kernel have finite dimension, then $V$ has finite dimension.
 A: Let $\{v_1,v_2,\ldots ,v_k\}$ be a basis of $\ker T$ and let $\{Tw_1,Tw_2,\ldots ,Tw_n\}$ form a basis of $W$.
Check that $\{v_1,v_2,\ldots ,v_k,w_1,w_2,\ldots ,w_n\}$ is a spanning set  of $V$ and hence $\dim V\le n+k\implies \dim V<\infty$
EDIT:
Let $x\in V$. If $Tx=0\implies x=\sum c_iv_i+\sum 0\cdot Tw_i$
If $Tx\neq 0\implies Tx=\sum d_iTw_i\implies x-\sum d_iw_i\in \ker T\implies x-\sum d_iw_i=\sum c_iv_i\implies x=\sum d_iw_i+\sum c_iv_i$
A: One of possible formulations of Rank–nullity theorem is that $$\dim(V)=\dim(\operatorname{Ker} T)+\dim(\operatorname{Im} T).$$
So if you know (or can prove) this result, you are done. 
Or, to be more precise, if you look at proof of this result, you can see that you get also that $V$ is finite dimensional if both $\operatorname{Ker} T$ and $\operatorname{Im} T)$ are. (This addition is mainly because - as pointed out in comments - rank-nullity theorem is usually formulated with the assumption that $V$ is finite dimensional.)
You can probably find a few posts about this on this site. Quick search lead me, for example, to: Proof that $\dim V = \dim \phi(V)+\dim \ker \phi$.
