$\newcommand{\im}{\operatorname{im}}$I am trying to prove or disprove the following statement:
Let $V$ and $W$ be finite-dimensional vector spaces. If $T:V\rightarrow W$ is a linear transformation then $V=\ker T\oplus \im T$. (By the symbol $\oplus$ I mean the direct sum of two vector spaces.)
This statement cannot be true if $V\neq W$ because a vector space can only be a direct sum of its subspaces. However, I am not sure about the case when $V=W$, i.e., when $T$ is a linear operator.
I want to use the following proposition:
$\textbf{Proposition.}$ Let $V$ be a finite-dimensional vector space and let $U$ and $W$ be subspaces of $V$. Then $V=U\oplus W$ if and only if $V=U+W$ and $U\cap W=\left\{ 0 \right\}$.
First I want to show that $V=\ker T + \im T$. I just don't have a clue how to possibly do this, which leads me to believe there must be a counterexample. I believe that $\ker T\cap \im T=\left\{0\right\}$ since $T(0)=0$ for any linear transformation and it is not possible for $Tv\neq 0$ if $v\in \ker T$. Some help?
Thank you in advance.
Best Answer
You have the right ideas. Indeed, your claim is not true. Consider, for example, the transformation $$ T = \pmatrix{0&1\\0&0} $$ Verify that im$(T) = \ker(T)$, and that both of these are one-dimensional subspaces of $\Bbb R^2$.
Notably, however, the statement will hold for any self-adjoint (symmetric) operator $T:V \to V$.