In Trefethen and Bau's book, Computational Linear Algebra, in the Projections chapter I've come across the following statement:

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> Let $S\_1$ and $S\_2$be two subspaces of $\mathbb{C}^m$ such that $S\_1 \cap S\_2 = \{0\} $ and $S\_1+S\_2=\mathbb{C}^m$ (where the sum denotes the span of $S\_1$ and $S\_2$. Then there exists a projector P such that $range(P)=S\_1$ and $null(P)=S\_2$
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They claimed that an equivalent problem is:

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> Given $v$, find vectors $v\_1 \in S\_1$, and $v\_2 \in S\_2$ such that $v\_1+v\_2=v$. 
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and the proof to the problem is:

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> The projection $Pv$ gives $v\_1$ and the complementary projection $(I-P)v$ gives $v2$
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1. My Questions: why are the problems equivalent, namely, why does the second problem suggest that there in fact exists a projection $P$?
2. how does their proof suggests that there exists a projection $P$?