range of projection is invariant under a linear operator

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Linear Algebra, 2nd Edition - Kenneth Hoffmann, Ray Kunze - Sec. 6.8 p. 218 Exercise 1

Let $E$ be a projection of $V$ and let $T$ be a linear operator on $V$. Prove that the range of $E$ is invariant under $T$ if and only if $ETE=TE$. Prove that both the range and null space of $E$ are invariant under $T$ if and only if $ET=TE$.

Czhang271828

Hint: Let $V=V_1\oplus V_2=:\binom{V_1}{V_2}$ such that $E(V)=\binom{V_1}{O}$. Whenever the range of $E$ is invariant under $T$, we have that
$$
ET\binom{V_1}{V_2}=\binom{T_{11}V_1+T_{12}V_2}{O}=\binom{T_{12}V_2}{T_{21}V_1}=TE\binom{V_1}{V_2}.
$$
Which is also equivalent to $T_{21}V_1=0$, that is, $ETE=TE$.

As for the second proposition, both the range and null space of $E$ are invariant under $T$ whenever $T_{11},T_{12}=0$.