# Linear Transformation

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.

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

#### Attachments

• math.gif
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shmoe
Homework Helper
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).

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
Homework Helper
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?

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

shmoe