How to Prove the Decomposition Theorem for Projections on a Vector Space?

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Here is this week's POTW:

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Let $P_1,\ldots, P_n$ be a sequence of projections on a vector space $V$ such that $P_iP_j = 0$ whenever $i \neq j$ and $P_1 + \cdots + P_n = I$. Prove that

$$V = \operatorname{Im}(P_1) \oplus \cdots \oplus \operatorname{Im}(P_n).$$

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Note: By a projection $P$ on a vector space $V$, I mean a linear operator on $V$ such that $P^2 = P$.

Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!

 

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No one answered this week's problem. You can find my solution below.

Spoiler

Given $v \in V$, $v = P_1(v)+ P_2(v) + \cdots + P_n(v)$. Furthermore, since the $P_i$ are projections, $P_i(v) \in \operatorname{Im}(P_i)$. Thus $v \in \operatorname{Im}(P_1) + \cdots + \operatorname{Im}(P_n)$ and consequently $V = \operatorname{Im}(P_1) + \cdots + \operatorname{Im}(P_n)$. Now suppose $0 = v_1 + \cdots + v_n$ for some $v_i \in \operatorname{Im}(P_i)$. For each $k\in \{1,2,\ldots, n\}$, $P_k(v_i) = 0$ for all $i \neq k$; this is due to the condition $P_iP_j = 0$ whenever $i \neq j$. So from the equation $0 = v_1 + \cdots + v_n$, we deduce $0 = P_k(v_k) = v_k$. It follows that $v_1 = \cdots = v_n = 0$. Hence, $V = \operatorname{Im}(P_1) \oplus \cdots \oplus \operatorname{Im}(P_n)$.