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Linear Algebra

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Moderate

Matrix of Rotation in Nonstandard Basis

LINALG-4QWIFV

Let $v_1=\begin{pmatrix}1\\\ 1\end{pmatrix}$, $v_2=\begin{pmatrix}1\\\ -1\end{pmatrix}$, and $\mathcal{B}=\{v_1,v_2\}$.

What is the matrix of the linear transformation which rotates vectors of $\mathbb{R}^2$ (counter-clockwise) by $\cfrac{\pi}{4}$ with respect to the basis $\mathcal{B}$?

A

$\begin{pmatrix}\cfrac{1}{\sqrt{2}}&-\cfrac{1}{\sqrt{2}}\\\ \cfrac{1}{\sqrt{2}}&\cfrac{1}{\sqrt{2}}\end{pmatrix}$

B

$\begin{pmatrix}\cfrac{1}{\sqrt{2}}&\cfrac{1}{\sqrt{2}}\\\ -\cfrac{1}{\sqrt{2}}&\cfrac{1}{\sqrt{2}}\end{pmatrix}$

C

$\begin{pmatrix}\cfrac{1}{\sqrt{2}}&0\\\ 0&\cfrac{1}{\sqrt{2}}\end{pmatrix}$

D

$\begin{pmatrix}0&\cfrac{1}{\sqrt{2}}\\\ \cfrac{1}{\sqrt{2}}&0\end{pmatrix}$

E

$\begin{pmatrix}0&-\cfrac{1}{\sqrt{2}}\\\ \cfrac{1}{\sqrt{2}}&0\end{pmatrix}$