How can I find the matrix of a linear transformation given a rotation in R^3?

[adc85]

OK I already have the answer for this problem but I don't know how my teacher came up with the answer:

Linear transformation T in R^3 consists of the rotation around x3 axis at the positive (counter-clockwise) direction at the angle 90 degrees. Such rotation transforms x1-axis into x2-axis. Find the matrix of this transformation.

Diagram for this problem is attached.

Also, the answer is:

[0 -1 0]
[1 0 0]
[0 0 1]

 

[shmoe]

You may have seen general form for rotations in three (or two) dimensions in class, but I'll describe something that will sometimes be usefull for more general linear transformations as well. The basic idea is once you know what the transformation does to a basis, you know the transformation.

Let X1, X2, X3 be the standard basis vectors, e.g. X1=[1,0,0]. Then T(X1) is the first column of the matrix corresponding to T (if you don't know this, you should try to prove it). You should know what T(X1) is by the description of the transformation. The second and third columns are found by considering T(X2) and T(X3).

 

[adc85]

OK I still don't really understand what is going on.

So you have these standard basis vectors. If we were to do no transformations then the transformation matrix would be:

[ 1 0 0 ]
[ 0 1 0 ]
[ 0 0 1 ]

Correct?

And based on that, X1 would now have the coordinates (0, 1, 0) making the matrix look like:

[0 0 0]
[1 1 0]
[0 0 1]

I don't see how the second column has the coordinates (-1, 0, 0) and how the third column stays the same.

 

[OlderDan]

OK I still don't really understand what is going on.

So you have these standard basis vectors. If we were to do no transformations then the transformation matrix would be:

[ 1 0 0 ]
[ 0 1 0 ]
[ 0 0 1 ]

Correct?

And based on that, X1 would now have the coordinates (0, 1, 0) making the matrix look like:

[0 0 0]
[1 1 0]
[0 0 1]

I don't see how the second column has the coordinates (-1, 0, 0) and how the third column stays the same.

A rotation that preserves X3 does more than rotate X1 into X2. What else has to happen?

 

[adc85]

Rotates X2 into ... X3? But why are the coordinates (-1, 0, 0) for that?

 

[shmoe]

If you can't visualize what will happen, building a physical model may help here. Attatch three sticks together at right angles to each other and label them x1, x2, x3 (I mean physically go do it). Rotate them with x3 as the axis and x1 goes to x2. Can you describe the new location of x2 in terms of the old basis?

 

[adc85]

If you can't visualize what will happen, building a physical model may help here. Attatch three sticks together at right angles to each other and label them x1, x2, x3 (I mean physically go do it). Rotate them with x3 as the axis and x1 goes to x2. Can you describe the new location of x2 in terms of the old basis?

Oooooh I see now! Thanks so much for your help. That helped tremendously. Now I see where he got the -1 one. It will be on the opposite side of where x1 used to be and x3 will stay in the same spot despite being rotated.