# Orthogonal matrices prove: T is orthogonal iff [T]_bb is an orthogonal matrix

1. Feb 22, 2010

### zeion

1. The problem statement, all variables and given/known data

Let B = {v1, ..., vn} be an arbitrary orthonormal basis of Rn, prove T is orthogonal iff $$[T]_{BB}$$ is an orthogonal matrix.

Hint: If B is orhtogonal basis for Rn then, $$x.y = [x]_B . [y]_B$$for all x, y in Rn.

3. The attempt at a solution

If $$[T]_{BB}$$ is an orthogonal matrix then

1) $$||[T(x)]_B|| = ||[x]_B||$$

2) $$[T(x)]_B . [T(y)]_B = [x]_B . [y]_B$$

and since B is orthonormal,

$$||[x]_B|| = ||x||$$

$$[x]_B . [y]_B = x.y$$

That's all I've got so far.. is this even right? How do I tie it into T being orthogonal?