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

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

I was given the hint to construct a basis for W, and use definition of direct sum to argue that W $\oplus$ W'=V. But I don't get the logic behind this.

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I don't think the question is very clear. If you want to project $W$ on $W'$, $W'$ needs to be a subspace of $W$, thus can't be the complement of $W$. If this is what the question is asking we can do so by constructing $W'$ and $T$ explicitly:

Choose any subspace $W'$ of $W$, fet $\{w_1,...,w_m\}$ be a basis of $W'$, extend this basis to a basis of $W$: $\{w_1,...w_m,w_{m+1},...w_{m+r}\}$, and extend further to a basis of V: $\{w_1,...w_m,w_{m+1},...w_{m+r},w_{m+r+1},...w_{m+r+s}\}$ where $\dim W' = m, \dim W = m+r, \dim V = m+r+s$. Now define $T(a_1w_1,...,a_{m+r+s}w_{m+r+s}) = a_1v_1 + ... + a_mw_m \in W'$. It's easy to check this satisfies the projection property.

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