Consider the linear transformation, $\displaystyle T:\mathbb{R}^2\rightarrow \mathbb{R}^2$, for which $\displaystyle T(x)$ is found by rotating the point x through an angle of $\displaystyle \frac{2\pi}{3}$ anticlockwise about the origin.

a) Find the 2 x 2 matrix A for which T(x) = Ax

I wasn't quite sure how $\displaystyle T(x)$ could = Ax and as such I wasn't entirely sure how to answer the question. I just applied the same method I had been using for all the previous question to get:

$\displaystyle \left(\begin{matrix} cos(\frac{2\pi}{3}) &-sin(\frac{2\pi}{3}) \\sin(\frac{2\pi}{3}) & cos(\frac{2\pi}{3})\end{matrix}\right)$

Is this the correct answer for matrix A? I thought this would be $\displaystyle T(x)$? What is the x value that A is multiplied by?

b) Write down the transformation that returns the point x to its original position.

Since we rotated everything $\displaystyle \frac{2\pi}{3}$ counter-clockwise, I would presume that to reverse that transformation we would rotate everything by $\displaystyle -\frac{2\pi}{3}$ to get:

$\displaystyle \left(\begin{matrix} cos(\frac{2\pi}{3}) & sin(\frac{2\pi}{3}) \\-sin(\frac{2\pi}{3}) & cos(\frac{2\pi}{3})\end{matrix}\right)$

Is this correct?

Thanks.