Exercise 4

Consider the vectors $$\vvec_1=\fivevec1{-1}{-1}11,\hspace{24pt} \vvec_2=\fivevec2{1}{4}{-4}2,\hspace{24pt} \vvec_3=\fivevec5{-4}{-3}71$$ and the subspace $W$ of $\real^5$ that they span.

1. Find an orthonormal basis for $W$.

2. Find the $5\times5$ matrix that projects vectors orthogonally onto $W$.

3. Find $\bhat$, the orthogonal projection of $\bvec=\fivevec{-8}3{-12}8{-4}$ onto $W$.

4. Express $\bhat$ as a linear combination of $\vvec_1$, $\vvec_2$, and $\vvec_3$.