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A linear transformation of C-vector space (complex field) where $T: V → V$ and $T ◦ T = −2T$.

$$\dim(V) = n$$

How can we prove that $\operatorname{Im}(T) + \ker(T)$ is direct? I know that i have to show that their intersection is $0$ and that $\operatorname{Im}(T) + \ker(T) = V$ but i can't seem to use the information they've given to me.

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Suppose $v \in Im(T) \cap ker(T)$ and $v \neq 0$. Then there exists a vector $w$ such that $Tw = v$ and $T^2w = Tv = 0$. But $T^2 = -2T$ thus $$0 = T^2w = -2Tw = -2v$$ implying $v = 0$, contradicting our hypothesis.

Hence $Im(T) \cap ker(T) = \{ 0 \}$

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