Exercise 1

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

1. Find the Gram matrix $G=A^TA$ and use it to find the singular values and right singular vectors of $A$.

2. Find the left singular vectors.

3. Form the matrices $U$, $\Sigma$, and $V$ and verify that $A=U\Sigma V^T$.

4. What is $\rank(A)$ and what does this say about $\col(A)$?

5. Determine an orthonormal basis for $\nul(A)$.