Linear algebra - Orthogonal matrix

[sweetiepi]

Homework Statement

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.

Homework Equations

None I don't think.

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.

 

[Rasalhague]

A transformation is orthogonal if its matrix representation obeys a certain equation. In an orthonormal basis, this equation reduces to T^-1=T^T. (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?