(a) Given $\mathbf{y} \in \mathbb{R}^{3}$ consider the linear transformation $T$ which maps

$$\mathbf{x} \mapsto T \mathbf{x}=\left(\mathbf{x} \cdot \mathbf{e}_{1}\right) \mathbf{e}_{1}+\mathbf{x} \times \mathbf{y}$$

Express $T$ as a matrix with respect to the standard basis $\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}$, and determine the rank and the dimension of the kernel of $T$ for the cases (i) $\mathbf{y}=c_{1} \mathbf{e}_{1}$, where $c_{1}$ is a fixed number, and (ii) $\mathbf{y} \cdot \mathbf{e}_{1}=0$.

(b) Given that the equation

$$A B \mathbf{x}=\mathbf{d}$$

where

$$A=\left(\begin{array}{ccc} 1 & 1 & 0 \\ 0 & 2 & 3 \\ 0 & 1 & 2 \end{array}\right), \quad B=\left(\begin{array}{ccc} 1 & 4 & 1 \\ -3 & -2 & 1 \\ 1 & -1 & -1 \end{array}\right) \quad \text { and } \quad \mathbf{d}=\left(\begin{array}{l} 1 \\ 1 \\ k \end{array}\right)$$

has a solution, show that $4 k=1$.