- #1
zeion
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Homework Statement
Let B = {v1, ..., vn} be an arbitrary orthonormal basis of Rn, prove T is orthogonal iff [tex][T]_{BB}[/tex] is an orthogonal matrix.
Hint: If B is orhtogonal basis for Rn then, [tex]x.y = [x]_B . [y]_B[/tex]for all x, y in Rn.
3. The Attempt at a Solution
If [tex][T]_{BB}[/tex] is an orthogonal matrix then
1) [tex] ||[T(x)]_B|| = ||[x]_B|| [/tex]
2) [tex] [T(x)]_B . [T(y)]_B = [x]_B . [y]_B[/tex]
and since B is orthonormal,
[tex] ||[x]_B|| = ||x||[/tex]
[tex][x]_B . [y]_B = x.y[/tex]That's all I've got so far.. is this even right? How do I tie it into T being orthogonal?