Exercise 4

Suppose that the matrix $A$ has the singular value decomposition $$\begin{bmatrix} -0.46 & 0.52 & 0.46 & 0.55 \\ -0.82 & 0.00 & -0.14 & -0.55 \\ -0.04 & 0.44 & -0.85 & 0.28 \\ -0.34 & -0.73 & -0.18 & 0.55 \end{bmatrix} \begin{bmatrix} 6.2 & 0.0 & 0.0 \\ 0.0 & 4.1 & 0.0 \\ 0.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 0.0 \end{bmatrix} \begin{bmatrix} -0.74 & 0.62 & -0.24 \\ 0.28 & 0.62 & 0.73 \\ -0.61 & -0.48 & 0.64 \end{bmatrix}.$$

1. What are the dimensions of $A$?

2. What is $\rank(A)$?

3. Find orthonormal bases for $\col(A)$, $\nul(A)$, $\col(A^T)$, and $\nul(A^T)$.

4. Find the orthogonal projection of $\bvec=\fourvec102{-1}$ onto $\col(A)$.