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To find the orthogonal projection of $v = (0,0,5,0)^T$ onto the subspace $W$ spanned by $(-1,-1,-1,1)^T,(-1,1,1,1)^T,(-1,-1,1,-1)^T$.

Since the vectors which forms a basis for $W$ are already orthogonal, the orthogonal projection of $v = (0,0,5,0)^T$ onto the subspace $W$ is the sum of the projection of $v$ on each of the basis vectors.

Is my logic and method correct?

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Yes, it's fine assuming that by projection of $v$ on a a vector $w$ you mean $\frac{\langle v,w\rangle}{\lVert w\rVert|^2}w$.

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