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A4_soln - c n d n and k ~x k 2 = c 2 1 c 2 n Solution...

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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. 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. Solution: We have that I = R T R so 1 = det I = det R T R = det R T det R = (det R ) 2 . Hence det R = ± 1. The matrix A = 3 2 1 1 has determinant 1, but is not orthogonal since the columns are not orthonormal. 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 Rx = x for every x R n . Solution: Suppose that R is orthogonal, then R T R = I . Then for any x R n we have Rx 2 = ( Rx ) · ( Rx ) = ( Rx ) T ( Rx ) = ( x T R T )( Rx ) = x T ( R T R ) x = x T x = x 2 . Hence Rx = x for every x .
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2 4. Let { v 1 , . . . , v n } be an orthonormal basis for an inner product space V with inner product , 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
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Unformatted text preview: + c n d n , and k ~x k 2 = c 2 1 + · · · + c 2 n . Solution: We have < ~x,~ y > = < c 1 ~v 1 + · · · + c n ~v n , d 1 ~v 1 + · · · + d n ~v n > = c 1 < ~v 1 , d 1 ~v 1 + · · · + d n ~v n > + · · · + c n < ~v n , d 1 ~v 1 + · · · + d n ~v n > = c 1 d 1 < ~v 1 ,~v 1 > + · · · + c 1 d n < ~v 1 ,~v n > + c 2 d 1 < ~v 2 ,~v 1 > + c 2 d 2 < ~v 2 ,~v 2 > + · · · + · · · + c 2 d n < ~v 2 ,~v n > + · · · + c n d 1 < ~v n ,~v 1 > + · · · + c n d n < ~v n ,~v n > But, since { ~v 1 , . . . ,~v n } is orthonormal we have < ~v i ,~v i > = 1 and < ~v i ,~v j > = 0 for i 6 = j , hence we get < ~x,~ y > = c 1 d 1 + · · · + c n d n , as required. Now observe that by taking d i = c i for all i we get y = x and obtain k ~x k 2 = < ~x,~x > = c 2 1 + · · · + c 2 n , as required....
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