Exercise 5

Consider the matrix $A = \begin{bmatrix} 1 & 0 & -1 \\ 2 & 2 & 0 \\ -1 & 1 & 2\\ \end{bmatrix}$.

1. Construct the Gram matrix $G$ and use it to find the singular values and right singular vectors $\vvec_1$, $\vvec_2$, and $\vvec_3$ of $A$. What are the matrices $\Sigma$ and $V$ in a singular value decomposition?

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

3. Find as many left singular vectors $\uvec_j$ as you can using the relationship $A\vvec_j=\sigma_j\uvec_j$.

4. Find an orthonormal basis for $\nul(A^T)$ and use it to construct the matrix $U$ so that $A=U\Sigma V^T$.

5. State an orthonormal basis for $\nul(A)$ and an orthonormal basis for $\col(A)$.