Finding the matrix of a transformation

[Cankur]

Homework Statement

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.

Homework Equations

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]

 

[Mark44]

Homework Statement

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.

Homework Equations

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.

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>?

So therefore, the matrix of the transformation should be:

[0 1]
[1 0]

 

[HallsofIvy]

Homework Statement

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.

Homework Equations

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]

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?

 

[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?

 

[Mark44]

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?

No, the matrix above doesn't work.

I'll ask this again.

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

 

[HallsofIvy]

Which (because I just have to jump in behind Mark44) is much what I asked before: what vector, <x, x>, has length 1? What vector <-x, x>, has length 1?