# Finding the matrix of a transformation

1. Oct 20, 2011

### Cankur

1. The problem statement, all variables and given/known data
Consider the transformation T from R2 to R2 that rotates any vector x through an angle of 45 degrees in the counterclockwise direction. You are told that T is a linear transformation. Find the matrix of T.

2. Relevant equations

3. The attempt at a solution

A vector with components x1 and x2 should become x 2 and x1, seeing that the transformation should rotate the vector 45 degrees. So therefore, the matrix of the transformation should be:

[0 1]
[1 0]

2. Oct 20, 2011

### Staff: Mentor

The rotation you're describing here is a rotation of 90° counterclockwise, not 45°.

What should T do to x1 = <1, 0> and x2 = <0, 1>?

3. Oct 20, 2011

### HallsofIvy

Staff Emeritus
No, with a 45 degree rotation, any vector on the x-axis (y=0) would rotate to the line y= x while any vector on the y-axis would rotate to the line y= -x. What vectors on those lines have length 1?

4. Oct 20, 2011

### Cankur

It should move it half of 90 degrees. But what matrix would achieve that? A matrix that looks like this perhaps:

[0 1/2]
[1/2 0]

How should you think when you have problems of this sort?

5. Oct 20, 2011

### Staff: Mentor

No, the matrix above doesn't work.