Let $f:\ \mathbb R^n\longrightarrow\mathbb R^n$ be a linear map. Moreover, $f$ is symmetric, which means $$f(x)\cdot y=x\cdot f(y),\ \forall x,y\in\mathbb R^n.$$ I want to show that
(a) $Ker\,f^\top\leq Im(f)$
(b) $Ker(f^2)\leq Ker(f)$
Could anyone give me some hints for this problem ? Thank you