Exercise 6

Consider the matrix $B=\begin{bmatrix} 1 & 0 \\ 2 & -1 \\ 1 & 2 \end{bmatrix}$ and notice that $B=A^T$ where $A$ is the matrix in this exercise.

1. Use your result from this exercise to find a singular value decomposition of $B=U\Sigma V^T$.

2. What is $\rank(B)$? Determine a basis for $\col(B)$ and $\col(B)^\perp$.

3. Suppose that $\bvec=\threevec{-3}47$. Use the bases you found in the previous part of this exercise to write $\bvec=\bhat+\bvec^\perp$, where $\bhat$ is in $\col(B)$ and $\bvec^\perp$ is in $\col(B)^\perp$.

4. Find the least-squares approximate solution to the equation $B\xvec=\bvec$.