A4_soln - Math 235 Assignment 4 Solutions 1. Prove that the...

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Assignment 4 Solutions 1. Prove that the product of two orthogonal matrices is an orthogonal matrix. Solution: Let P and Q be orthogonal matrices. Then we have ( PQ ) T ( PQ ) = Q T P T PQ = Q T Q = I, since P T P = I and Q T Q = I . Thus PQ is also orthogonal. 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 . Solution: Suppose that R is orthogonal, then R T R = I . Then for any ~x R n we have k R~x k 2 = ( R~x ) · ( R~x ) = ( R~x ) T ( R~x ) = ( ~x T R T )( R~x ) = ~x T ( R T R ) ~x = ~x T ~x = k ~x k 2 . Hence k R~x k = k ~x k for every ~x . 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 = h ~v i ,~v j i a) Prove that G is symmetric ( G T = G ). Solution: Because the inner product is symmeric,
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This note was uploaded on 11/30/2010 for the course MATH 235/237 taught by Professor Wilkie during the Spring '10 term at Waterloo.

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A4_soln - Math 235 Assignment 4 Solutions 1. Prove that the...

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