Let $f$ : $\mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ and $g$ : $\mathbb{R}^{m} \rightarrow \mathbb{R}^{n}$ linear functions such that $(g \circ f)(x) = x \hspace{0.4cm} \forall x \in \mathbb{R}^{n}$
How can I prove that:
$\mathbb{R}^{n} = Kernel(f) \oplus Image(g)$ where $\oplus$ is the direct sum.
So far i wrote this:
$\mathbb{R}^{n} = Kernel(f) \oplus Kernel(f)^{\perp}$
So if I can prove that $Kernel(f)^{\perp} = Image(g)$ I'm done. But I'm not so sure how!
Thank's a lot!