# A4 - = ± x 1 x 2 x 3 ² G y 1 y 2 y 3 c Consider the inner...

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Math 235 Assignment 4 Due: Wednesday, Oct 13th 1. Prove that the product of two orthogonal matrices is an orthogonal matrix. 2. 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 k R~x k = k ~x k for every ~x R n . 3. Suppose that B = { ~v 1 ,~v 2 ,~v 3 } is a basis for a real inner product space V . Deﬁne a 3 × 3 matrix G by ( G ) ij = < ~v i ,~v j > a) Prove that G is symmetric ( G T = G ). b) Show that if [ ~x ] B = x 1 x 2 x 3 and [ ~ y ] B = y 1 y 2 y 3 , then < ~x,~ y >
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Unformatted text preview: = ± x 1 x 2 x 3 ² G y 1 y 2 y 3 c) Consider the inner product h , i for P 2 deﬁned by h p,q i = p (0) q (0) + p (1) q (1) + p (2) q (2). Determine the matrix G of this inner product with respect to the basis { 1 ,x,x 2 } . 4. Let V be a real inner product space with inner product h , i and let ~u,~v ∈ V . Prove that k ~u + ~v k 2 = k ~u k 2 + k ~v k 2 if and only if h ~u,~v i = 0....
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