Question

Find the standard matrix for the stated composition in R^2.
a) A rotation of 90, followed by a reflection about the line y=x.
b) An orthogonal projection on the y-axis, followed by a contraction with factor k=1/2.
c) A reflection about the x-axis, followed by a dilation with factor k=3.

Answer 1

what is the matrix of rotation of 90?

Answer 2

that is
\[\left[\begin{matrix}0&-1\\1&0\end{matrix}\right]\]

Answer 3

it goes from \[\left[\begin{matrix}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{matrix}\right]\]
to\[\left[\begin{matrix}\sin \theta & -\cos \theta \\ \cos \theta & \sin \theta\end{matrix}\right]\]

Answer 4

now, the matrix to reflect about the line y =x is
\[\left[\begin{matrix}0&1\\1&0\end{matrix}\right]\]

Answer 5

the same

Answer 6

but I use numbers to easily times them together

Answer 7

for a) rotation first, then reflection about y=x, so that the second matrix is written first, then the rotation matrix, that is
\(\left[\begin{matrix}0&1\\1&0\end{matrix}\right]\)*\(\left[\begin{matrix}0&-1\\1&0\end{matrix}\right]\)

Answer 8

now, you compute the result

Answer 9

oh, thanks, i had them reversed at first....

Answer 10

:)

Answer 11

can you assist with part b?