# Show that a vectorial subspace $W$ invariant under a linear transformation $T$ is the kernel of $T$.

Let $V$ a finite vectorial space and $T:V\to V$ a linear transformation. Show that if $W$ is a subspace of $V$ invariant under $T$ so that $V={\rm{Im}}(T)\oplus W$ then $W=\ker T$.

• Have you tried anything? – Brian Fitzpatrick Jul 11 '17 at 23:54
• I tried with the range-nulity theorem but I not Have exit! – Diego1802 Jul 11 '17 at 23:56
• for any $w\in W$, $w\in Im(T)$ by definition. But $W$ is $T$ invariant, what does it mean? – chan kifung Jul 11 '17 at 23:58