Question: Problem 5. Let T: R -> R be an orthogonal linear transformation. (a) Show that T preserves angles. That is, prove that for all nonzero

Problem 5. Let T: R" -> R" be an orthogonal linear transformation. (a) Show that T preserves angles. That is, prove that for all nonzero vectors U, we R", if the angle between 7 and w is 0, then the angle between T(7) and T(w) is also 0. [HINT: See Definition 5.1.12.] (b) Conversely, if T: R" - R" is a linear transformation that preserves angles, is T necessarily orthogonal? Prove your claim. Problem 6. Let A be an m x n matrix with linearly independent columns, and let b e Rm. Suppose A = QR is the QR-factorization of A. Show that the system of linear equations AT = b has a unique least squares solution * = R-'QTb

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