# Matrix transformation 2

• Nov 23rd 2011, 03:40 AM
terrorsquid
Matrix transformation 2
Consider the linear transformation, $T:\mathbb{R}^2\rightarrow \mathbb{R}^2$, for which $T(x)$ is found by first rotating the point $x$ through an angle of $\frac{\pi}{3}$ anticlockwise about the origin and then reflecting that point across the line $y = x$.

Find the 2x2 matrix A for which $T(x) = Ax$

I understand how to transform matrices in the required ways described above, what I don't understand is which matrix I am transforming to begin with, i.e., to rotate a matrix anticlockwise bt \frac{\pi}{3} I just multiply it by:

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

Similarly, to reflect a matrix across the line y = x I just multiply it by:

$\left(\begin{matrix} 0 &1\\1 & 0\end{matrix}\right)$

What I don't understand in this question is which matrix I am transforming in the first place? What does rotating the point x mean? What matrix am I transforming by multiplying it by:

$\left(\begin{matrix} cos(\frac{\pi}{3}) &-sin(\frac{\pi}{3}) \\sin(\frac{\pi}{3}) & cos(\frac{\pi}{3})\end{matrix}\right)$ $\left(\begin{matrix} 0 &1\\1 & 0\end{matrix}\right) =$ $\left(\begin{matrix} -sin(\frac{\pi}{3}) &cos(\frac{\pi}{3}) \\cos(\frac{\pi}{3}) & sin(\frac{\pi}{3})\end{matrix}\right)$

Which matrix is A and which is x? do I need to multiply everything by the column vector $(x_1,x_2)$ or something?

Thanks.
• Nov 23rd 2011, 06:35 AM
Deveno
Re: Matrix transformation 2
you have the matrix multiplication backwards (the first matrix you compose by goes on the RIGHT).

yes, you let the matrix product operate on the column vector (x,y) (or (x1,x2) if you prefer). the matrix A is the product of the two 2x2 matrices that each transformation is represented by.