Find Matrix A for 135° Clockwise Rotation in R^2

[UrbanXrisis]

Find the matrix A of the linear transformation T from $$R^2$$ to $$R^2$$ that rotates any vector through an angle of $$135^o$$ 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 don't 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 don't 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 don't even know how linear transformations have to do with rotations, not sure at all what is going on because this homework 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]

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

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 $$-\frac{\sqrt{2}}{2}$$ 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.