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My book has an exercise:

"Suppose that $W$ is a subspace of a finite-dimensional vector space $V$.

a) Prove that there exists a subspace $W'$ and a function $T:V\longrightarrow V$ such that $T$ is a projection on $W$ along $W'$

b) Give an example of a subspace $W$ of a vector space $V$ such that there are two projections on W along two (distinct) subspaces."

Do I understand this right, that in a) $W'$ is supposed to be a subspace of $V$, not of $W$, and that in b) two distinct subspaces of $V$, not $W$ are meant?

Thank you

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I assume that you have defined "$T\colon V\to V$ is a projection on $W$ along $W'$" to mean that $T(x)=x$ for all $x\in W$ and $T(x)=0$ for $x\in W'$ and $V=W+W'$. From that it should be clear that $W'$ will have to be a subspace of $V$. And in b) yo are to exhibit a single $V$ and a single $W$ and two different possible $W'$. – Hagen von Eitzen Oct 18 '12 at 15:00
yes, that's what I defined (in the exercise before), and that's exactly what I thought. I just met this kind of mentioning 'a subspace' without saying 'of what' a number of times, and it confused me every time. That's why I decided to ask here.. – foaly Oct 18 '12 at 15:05
up vote 0 down vote accepted

$W'$ is a subspace of $V$, since $W+W'=V$, $T(x)=0 \forall x \in W'$, $T(x)=x \forall x \in W$.

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