TITLE: $V = \operatorname{Im} T + \ker T $ then $ \operatorname{Im} T \cap \ker T = \{0\}$
QUESTION [5 upvotes]: Let $F$ be a field, let $V$ be a vector space with finite dimension over $F$ and let $T$ be a linear operator on $V$. Prove that:
a) If $V = \operatorname{Im} T + \ker T $ then $\operatorname{Im} T \cap \ker T = \{0\}$;
b) If $\operatorname{Im} T \cap \ker T = \{0\}$ then $V = \operatorname{Im} T \bigoplus \ker T $
I'm really stuck with this problem, some help to solve this please.

REPLY [2 votes]: These results hold only in finite dimensional vector spaces, so you must use a dimension argument somewhere. The rank-nullity theorem tells you that $\dim(V)=\operatorname{rk}(T)+\dim(\ker T)$, where $\operatorname{rk}(T)$ is by definition the dimension of the image of $T$. This reduces the question to the following (after which you should forget $T$)

If $U,W$ are subspaces of a space $V$ of finite dimension$~n$, and if $\dim U+\dim W=n$ then show that

if $U+W=V$ then $U\cap W=\{0\}$, and

if $U\cap W=\{0\}$ then $U\oplus W=V$.



If you know that in finite dimension one has $\dim(U+W)=\dim U+\dim W-\dim(U\cap W)$ for any subspaces $U,W$ (a formula notable for not extending in the obvious way to more than $2$ subspaces) then things become easy. For point 1.  the formula gives you $\dim(U\cap W)=n-(\dim U+\dim W)=0$ which allows you to conclude, and for point 2. it similarly gives you that $\dim(U+W)=\dim U+\dim W=n$.
In case you need a proof for the formula, the basic method is to choose a basis for $U\cap W$, extend this basis separately to bases of $U$ and of $W$, and then prove that the $\dim U+\dim W-\dim(U\cap W)$ vectors in the union of those bases together form a basis of $V$; this proof is easy though not trivial. A more slick proof consists of forming the vector space $X=U\times W=\{\,(u,w)\mid u\in U,w\in W\,\}$, which has $\dim X=\dim U+\dim V$, and then apply the rank-nullity theorem to $f: X\to V$ defined by $f(u,w)=u-w$, using that $\ker(f)=\{\, (v,v)\mid v\in U\cap W\,\}$ whence $\dim(\ker f)=\dim U\cap W$.