clockwise direction

UrbanXrisis

Find the matrix A of the linear transformation T from [tex]R^2[/tex] to [tex]R^2[/tex] that rotates any vector through an angle of [tex]135^o[/tex] in the clockwise direction.

my book does not talk about how to answer this question. I've seen a change in 90 degrees, but I dont know how to do a 135 degree.

Corneo

Look at the idea how to do it 90 degrees, the same idea applies.

UrbanXrisis

I guessed on how to get the 90 degree one, since it was mulitple choice. so i dont actually know the process, could someone explain it to me?

shmoe

You can look at what the transformation does to the standard basis, what is T(1,0) and T(0,1)? These determine the 1st and 2nd columns of A respectively and can be found using a little trig. More generally you can find a rotation by any angle this way.

UrbanXrisis

how can this be found with trig? i dont even know how linear transformations have to do with rotations, not sure at all what is going on because this hw questions came out of the blue

shmoe

I mean you can find T(1,0) and T(0,1) in terms of sin's and cos's of your angle. Can you find T(1,0) and T(0,1)? Drawing a picture will help.

A rotation about the origin is a linear transformation.

UrbanXrisis

so T(1,0) represents sin(90) and T(0,1) = cos(90)?

shmoe

No, T(1,0) is a vector. Did you draw a picture? Start with the vector (1,0). Rotate it 135 degrees clockwise. What quadrant is it in? What angle does it make with the x-axis? What are it's new coordinates?

UrbanXrisis

it's in the thrid quadrant. it makes a 45 degree angle with the x axis. so the x is [tex]-\frac{\sqrt{2}}{2}[/tex] which would be the same for the y

shmoe

Right, so that's the first column of the matrix for T. The second column is T(0,1), which you can find the same way.