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# A4 - k R~x k = k ~x k for every ~x ∈ R n 4 Let ~v 1,~v n...

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Math 235 Assignment 4 Due: Wednesday, Jun 2nd 1. Prove that the product of two orthogonal matrices is an orthogonal matrix. 2. Prove that if R is an orthogonal matrix, then det R = ± 1. Give an example of a matrix A that has det A = 1, but is not orthogonal. 3. Observe that the dot product of two vectors ~x,~ y R n can be written as ~x · ~ y = ~x T ~ y. Use this fact to prove that if an n × n matrix R is orthogonal, then
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Unformatted text preview: k R~x k = k ~x k for every ~x ∈ R n . 4. Let { ~v 1 , . . . ,~v n } be an orthonormal basis for an inner product space V with inner product h , i and let ~x = c 1 ~v 1 + · · · + c n ~v n and ~ y = d 1 ~v 1 + · · · + d n ~v n . Show that < ~x,~ y > = c 1 d 1 + · · · + c n d n , and k ~x k 2 = c 2 1 + · · · + c 2 n ....
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