Suppose $T:V\to V$ is a linear transformation, how t0 show $U+W$ and $U\cap W$ are invariant under $T$ if $U$ and $W$ are invariant under $T$ ?
My try:
$1.$ Let $u\in U,w\in W$. Then $T(u+w)=T(u)+T(w)$. Since $T(u)\in U,T(w)\in W$, so $T(u+w)\in U+W$. So $T(U+W)\subset U+W$
$2.$ Let $x\in U\cap W$. Then $x\in U$, then $T(x)\in U$. Also $x\in W$ and then $T(x)\in W$. Therefore $T(x)\in U\cap W$. Therefore $T(U\cap W)\subset U\cap W$
I feel my proof too short and a incomplete. Could someone edit?