GO Classes DA 2025 | Weekly Quiz 6 | Change of Basis & Linear Transformation | Question: 10

Question

Let $T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{2}$ be a linear transformation such that $T\left[\begin{array}{l}0 \\ 0 \\ 1\end{array}\right]=\left[\begin{array}{l}2 \\ 6\end{array}\right], T\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]=\left[\begin{array}{l}1 \\ 2\end{array}\right]$ , and $T\left[\begin{array}{l}2 \\ 1 \\ 3\end{array}\right]=\left[\begin{array}{c}7 \\ 20\end{array}\right]$. 
Which of the following is the standard matrix $A$ of $T$?

• A. $\left(\begin{array}{lll}1 & -1 & 2 \\ 2 & -2 & 6\end{array}\right)$
• B. $\left(\begin{array}{ccc}1 & 2 & 3 \\ -1 & -2 & -3\end{array}\right)$
• C. $\left(\begin{array}{lll}1 & 1 & 2 \\ 2 & 2 & 6\end{array}\right)$
• D. $\left(\begin{array}{lll}2 & 1 & 3 \\ 6 & 3 & 9\end{array}\right)$

Answer 1

T(x) = Ax where x is the input for the transformation

In the question above, x and T(x) are given, we have to find A

$ T(x) = \begin{bmatrix}
2 & 6 \\
1 & 2 \\
7 & 20
\end{bmatrix}$

$ x = \begin{bmatrix}
0 & 1 & 2 \\
0 & 0 & 1 \\
1 & 0 & 3
\end{bmatrix}$

Since |x| = 1, therefore x is invertible

$T(x) = Ax$
$A = T(x).x^{-1}$

Solving this will get the A = Option A