Maths 2 End Term: 30 April 2023, Set QPF1-S2 (January 2023 term)

Let $D$ denote the set of all $2 \times 2$ diagonal matrices. Consider an ordered basis $\beta = \left\{\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 \\ 0 & 2 \end{pmatrix}\right\}$. Let $T : U \to \mathbb{R}^2$ be a linear transformation defined as $T\begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix} = (a + b, 4a - 5b)$. Let matrix $A$ be the matrix representation of $T$ with respect to the ordered basis $\beta$ for $D$ and the standard ordered basis for the co-domain $\mathbb{R}^2$.

Answer the given subquestions.

Which of the following matrix is A?

Let $D$ denote the set of all $2 \times 2$ diagonal matrices. Consider an ordered basis $\beta = \left\{\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 \\ 0 & 2 \end{pmatrix}\right\}$. Let $T : U \to \mathbb{R}^2$ be a linear transformation defined as $T\begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix} = (a + b, 4a - 5b)$. Let matrix $A$ be the matrix representation of $T$ with respect to the ordered basis $\beta$ for $D$ and the standard ordered basis for the co-domain $\mathbb{R}^2$.

Answer the given subquestions.

Which of the following is/are true about T?