ISI PCA 2024 | Question : 2

Question

Let $\mathbf{C} = \{ (1, 2), (2, 1) \}$ be a basis of $\mathbb{R}^2$ and $T: \mathbb{R}^2 \to \mathbb{R}^2$ be defined by
\[
T \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} x + y \\ x - 2y \end{pmatrix}.
\]
If $T(C)$ represents the matrix of $T$ with respect to the basis $\mathbf{C}$, then which of the following is true?

 

1. $T(C) = \begin{pmatrix} -3 & -2 \\ 3 & 1 \end{pmatrix}$

2. $T(C) = \begin{pmatrix} 3 & -2 \\ -3 & 1 \end{pmatrix}$

3. $T(C) = \begin{pmatrix} -3 & -1 \\ 3 & 2 \end{pmatrix}$  

4. $T(C) = \begin{pmatrix} 3 & -1 \\ -3 & 2 \end{pmatrix}$

 

Comments

$ Option : C $ is correct. But if someone knows how,  then please explain.

---

explained...please go through the topics of change of basis and linear transformation

---

One more way to find A is by T(x) = A*x, for 2 values of x, given in the basis vecor C.. (1,2) and (2,1)...

Then following the equation T(x) = C^(-1) * A * C... for the basis C...
3rd option is correct..

Answer 1

 

$Correct$ $answer$ $will$ $be$ $option$ $C$ .
In case u want to my interview experience at IIT M : IIT M MS 2024 - GATE Overflow for GATE CSE .It contains the source to study these topics as well!

Answer 2