Exercise 1

Given a column vector $$\mathbf x = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix},$$ we define the transpose of $\mathbf x$ to be $$\mathbf x^T = \begin{pmatrix} x_1 & x_2 \end{pmatrix}.$$ If $$A = \begin{pmatrix} 3 & -2 \\ 0 & -1 \end{pmatrix}, \mathbf x = \begin{pmatrix} 4 \\ 1 \end{pmatrix}, \quad \text{and} \quad \mathbf y = \begin{pmatrix} -2 \\ 3 \end{pmatrix},$$ find each of the following.

1. $A \mathbf x$

2. $A \mathbf y$

3. $\mathbf x^T \mathbf y$

4. $\mathbf y^T \mathbf x$