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

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.

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.