Exercise 1

Suppose that $$\wvec_1=\threevec111,\hspace{24pt} \wvec_2=\threevec1{-2}1.$$

1. Verify that $\wvec_1$ and $\wvec_2$ form an orthogonal basis for a plane $W$ in $\real^3$.

2. Use Projection formula to find $\bhat$, the orthogonal projection of $\bvec=\threevec21{-1}$ onto $W$.

3. Find an orthonormal basis $\uvec_1$, $\uvec_2$ for $W$.

4. Find the matrix $P$ representing the matrix transformation that projects vectors in $\real^3$ orthogonally onto $W$. Verify that $\bhat = P\bvec$.

5. Determine $\rank(P)$ and explain its geometric significance.