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

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Math 235 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, ( G ) ji = h ~v j ,~v i i = h ~v i ,~v j i = ( G ) ij
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2 b) Show that if [ ~x ] B = x 1 x 2 x 3 and [ ~ y ] B = y 1 y 2 y 3 , then h ~x, ~ y i = x 1 x 2 x 3 G y 1 y 2 y 3 Solution: By the bilinearity of h , i h ~x, ~ y i = h x 1 ~v 1 + x 2 ~v 2 + x 3 ~v 3 , y 1 ~v 1 + y 2 ~v 2 + y 3 ~v 3 i = x 1 y 1 h ~v 1 ,~v 1 i + x 1 y 2 h ~v 1 ,~v 2 i + · · · + x 3 y 3 ~v 3 i = ( x 1 ( G ) 11 + x 2 ( G ) 21 + x 3 ( G ) 31 ) y 1 + ( x 1 ( G ) 12 + x 2 ( G ) 22 + x 3 ( G ) 32 ) y 2 + + ( x 1 ( G ) 13 + x 2 ( G ) 23 + x 3 ( G ) 33 ) y 3 = x 1 x 2 x 3 G y 1 y 2 y 3 c) Consider the inner product
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