|Apr14-10, 11:08 PM||#1|
Linear algebra - Orthogonal matrix
1. The problem statement, all variables and given/known data
Let T: Rn -> Rn be a linear transformation, and let B be an orthonormal basis for R^n. Prove that [ the length of T(x) ] = [ the length of x ] if and only if [T]B (the B-matrix for T) is an orthogonal matrix.
2. Relevant equations
None I don't think.
3. The attempt at a solution
I've always had a hard time grasping linear transformation problems, so I don't even really know where to start with this one.
|Apr15-10, 01:27 AM||#2|
A transformation is orthogonal if its matrix representation obeys a certain equation. In an orthonormal basis, this equation reduces to T-1=TT. (It's inverse is equal to its transpose.) So that's one relevant equation. How would you express, as a matrix equation, the fact that the length of a vector is unchanged by a transformation? Hint: if the length stays the same, what can we say about the square of the length?
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