GO Classes DA 2025 | Weekly Quiz 6 | Change of Basis & Linear Transformation | Question: 5

Question

Consider the linear map $T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ defined by
$$
T(x, y)=(x-y, x-2 y), \text { for } x, y \in \mathbb{R}
$$
Let $\mathcal{E}$ be the standard basis for $\mathbb{R}^{2}$ and let $\mathcal{B}=\{(3,1),(2,1)\}$ be another basis for $\mathbb{R}^{2}$.
Which of the following is the matrix for $T$ relative to the basis $\mathcal{B}$ ?

• A. $\left(\begin{array}{cc}0 & 1 \\ 1 & -1\end{array}\right)$
• B. $\left(\begin{array}{cc}1 & 1 \\ -1 & -1\end{array}\right)$
• C. $\left(\begin{array}{cc}1 & 0 \\ 0 & -1\end{array}\right)$
• D. $\left(\begin{array}{ll}0 & -1 \\ 1 & -1\end{array}\right)$

Answer 1

Answer: A

Answer 2

T matrix is $\begin{bmatrix} 1 & -1 \\ 1 & -2 \end{bmatrix}$ with respect to standard Basis

With respect to new Basis B

$[T]_{B} = B^{-1}TB$

and B is $\begin{bmatrix} 3 & 2 \\ 1 & 1 \end{bmatrix}$

B inverse = $\begin{bmatrix} 1 & -2 \\ -1 & 3 \end{bmatrix}$

Apply the formula, that is

$\begin{bmatrix} 1 & -2 \\ -1 & 3 \end{bmatrix}$ $\begin{bmatrix} 1 & -1 \\ 1 & -2 \end{bmatrix}$ $\begin{bmatrix} 3 & 2 \\ 1 & 1 \end{bmatrix}$

Answer will be $\begin{bmatrix} 0 & 1 \\ 1 & -1 \end{bmatrix}$ which is Option A