I have a problem with the following exercise:
Find the projection of a vector $v=[-2, 0, 2, 2, 0]^{T}$ onto subspace $V = span\{[-1, 1, 2, 3, 4 ]^{T}, [0, 0, 1, 2, 3]^{T}\}$ along subspace $W = span\{[1, 1, 1, 1, 1]^{T}, [0, 0, 3, 2, 1]^{T}, [0, 0, 1, 0, 1]^{T}\}$.
To be honest I don't know what exactly should I do. Could you give me some advice?
We are looking for a vector $v'\in V$ such that $v-v'\in W$.
So, first you should confirm that $V\oplus W=\Bbb R^5$, which follows if the 5 given generator elements form a basis $B$.
Then find the unique decomposition $v=v'+w$ with $v'\in V,\, w\in W$. (This becomes easy if you find the coordinates of $v$ with respect to basis $B$.)