Let $V$ be a vector space, $W$ a subspace of $V$ and $T : V \to V$ a linear operator. Suppose $V = \text{Im } T \oplus W$ and $W$ is $T$-invariant. I have to prove that
(a) $W \subseteq \text{Ker }T$, where $\text{Ker }T$ is the kernel of $T$.
(b) If $V$ is finite dimensional, then $W = \text{Ker }T$.
For (a), I take $w \in W$, and I have prove that $T(w)=0$. Since $W$ is $T$-invariant, then $T(w) \in W$. I really don't know what's next, because I don't know how to use the fact that $V = \text{Im } T \oplus W$. Can I say $W = \text{Im } T \oplus V$ from this fact? Also, I don't have any idea of how to prove (b). Any hints or ideas will be very appreciated. Thanks.
Best Answer
a) Since $V = \operatorname{im} T \oplus W$, then $\operatorname{im} T \cap W = \{0\}$. To see why, let $v \in \operatorname{im} T \cap W$. We have $v = v + 0 = 0 + v$. $v$ can be uniquely written as the sum of two vectors in each of $\operatorname{im} T$ and $W$. Therefore we must have $v = 0$. Now let $w \in W$. Since $T w \in W$ and $T w \in \operatorname{im} T$, it follows that $T w = 0$ and $w \in \ker T$.
b) We have $\dim V = \dim \operatorname{im} T + \dim \ker T = \dim \operatorname{im} T + \dim W$. Thus $\dim \ker T = \dim W$. Since $W \subset \ker T$, it follows that $\ker T = W$.