True or False?

Algebra Level 4

If \(A\) is any invertible \(2\times 2\) matrix, then there must exist two perpendicular unit vectors \(v\) and \(w\) such that the vectors \(Av\) and \(Aw\) are perpendicular as well.

Bonus Question: Can you generalize to 3 or even \(n\) dimensions?

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