Exercise 3.B.2

Question

Suppose V is a vector space and $S, T \in \mathcal{L}(V,V)$ are such that
$$
	ext{range } S \subset 	ext{null } T.
$$
Prove that $(ST)^2 = 0$.

Arden Shi · Accepted Answer

$$
(ST)^2
$$
Is equivalent to 
$$
STST
$$
First off, observe that 
$$
ST \subset 	ext{range } S \subset 	ext{null } T
$$
So, because $ST \subset 	ext{null } T$,
$$
TST = 0
$$
Thus, 
$$
STST = S(0) = 0
$$
\qed

Exercise 3.B.2

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Exercise 3.B.2

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