# Standard matrix of Rotation and Reflection

I have been fuzzy on this transformation stuff and am practicing for an upcoming test.

 Find the standard matrix: counterclockwise rotation through 60 degrees, followed by a reflection in the line y=x


I am not exactly sure how to do this. My textbook gives an example of how to find the standard matrix of a rotation by a formula. I also know that the standard matrix of a reflection of the line y=x is {{0,1},{1,0}}. But, I don't really know how to tackle this problem.

Thanks

## 1 Answer

Hint: The column vectors of the linear transformation matrix will be wherever the standard basis vectors are mapped to.

• Hmm, I don't know how I should put that statement in action – A A Apr 6 '14 at 1:14
• where the standard basis vectors are mapped to. What does this mean? – A A Apr 6 '14 at 1:17
• The vector $<1, 0>$, our first standard basis vector, gets rotated $60$ degrees, so it becomes $<.5, \frac{\sqrt{3}}{2}>$. Next, it gets reflected over the line $y=x$. In other words, the $x$ coordinate becomes the $y$-coordinate and vice versa. So we get $<\frac{\sqrt{3}}{2}, .5>$. This is therefore the first column vector in our matrix. For the second column in our matrix, you'll have to find where the second standard basis vector, $<0, 1>$, gets mapped to. – Kaj Hansen Apr 6 '14 at 1:17
• So the other column vector should be <0.5, sqrt(3)/2>, right? – A A Apr 6 '14 at 1:22
• I have used your method and the matrix should be {{sqrt(3)/2,0.5},{0.5,sqrt(3)/2}} – A A Apr 6 '14 at 1:27