Find an orthogonal basis for \(\mathcal{S}\) and use it to find the \(3 \times 3\) matrix \(P\) that projects vectors orthogonally into \(\mathcal{S}\).

Question

Let \(\mathcal{S} \subset \mathbb{R}^{3}\) be the subspace spanned by the two vectors \(v_{1}=(1,-1,0)\) and \(v_{2}=\) \((1,-1,1)\) and let \(\mathcal{T}\) be the orthogonal complement of \(\mathcal{S}\) (so \(\mathcal{T}\) consists of all the vectors orthogonal to \(\mathcal{S}\) ).

a) Find an orthogonal basis for \(\mathcal{S}\) and use it to find the \(3 \times 3\) matrix \(P\) that projects vectors orthogonally into \(\mathcal{S}\).

b) Find an orthogonal basis for \(\mathcal{T}\) and use it to find the \(3 \times 3\) matrix \(Q\) that projects vectors orthogonally into \(\mathcal{T}\).

c) Verify that \(P=I-Q\). How could you have seen this in advance?