## Acalytica

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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?
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