# Transformation Matrix

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• Jun 21st 2009, 05:04 PM
Fel
Transformation Matrix
Consider rotation of the plane R^2 using the origin as the pivot.

Let L: R^2 --> R^2 be rotation by an angle theta. Find the matrix for L with respect to the standard basis {(1,0),(0,1)}.

Thanks for any help!
• Jun 22nd 2009, 03:15 AM
HallsofIvy
The problem with you not showing any work is that we don't have any idea what you have to work with? Do you know that a rotation matrix must have determinant 1? And that the "length" of each column (thought of as a vector) must be 1?

If you know that, this problem is almost trivial. If you don't know it, do you know that you can find the matrix corresponding to a linear tranformation (in a given basis) by seeing what it does to the basis vectors? The coefficients of the result, as a linear combination of the basis vectors, form the columns of the matrices. With a rotation of $\displaystyle \theta$, what does the unit vector (1, 0) rotate to? Its x and y components form the first column of the matrix. Do the same thing with (0, 1).