Problem 1. Let $T_1(x_1, x_2, x_3) = (x_1 + x_2, x_1 - x_2, x_3)$ and $T_2(x_1, x_2, x_3) = (x_1, x_2 + x_3, x_3)$.
(a) Find the standard matrix for $T_1$ and $T_2$.
(b) Find the standard matrix for $T_1 \circ T_2$ and $T_2 \circ T_1$.
(c) Use the matrices obtained in part (b) to find formulas for $T_1(T_2(x_1, x_2, x_3))$ and $T_2(T_1(x_1, x_2, x_3))$.
[10 marks]
Problem 2. Find the standard matrix for the matrix operator defined by the equation and determine whether the operator is one-to-one.
$w_1 = x_1 + 2x_2 - x_3$
$w_2 = x_1 - 4x_2 + x_3$
$w_3 = x_1 - 3x_2 + x_3$
[10 marks]
Problem 3. Find $T^{-1}(w_1, w_2, w_3)$ for a given matrix
$w_1 = x_1 + 2x_2 + x_3$
$w_2 = -2x_1 + x_2 + 4x_3$
$w_3 = 7x_1 + 4x_2 - 5x_3$.
[10 marks]
Problem 4. Let $T_A : \mathbb{R}^3 \to \mathbb{R}^3$ be multiplication by
$A = \begin{bmatrix} 4 & -1 & 0 \\ 1 & 3 & 1 \\ -1 & 1 & 4 \end{bmatrix}$
and let $e_1, e_2$, and $e_3$ be the standard basis vectors for $\mathbb{R}^3$.
Find the following:
(a) $T_A(e_1), T_A(e_2), T_A(3e_3)$,
(b) $T_A(e_1 + e_2 + e_3)$.
[Total: 40 marks]
