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homework_11_solutions

# homework_11_solutions - MA 35100 HOMEWORK ASSIGNMENT#11...

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MA 35100 HOMEWORK ASSIGNMENT #11 SOLUTIONS Problem 1. pg. 218; prob. 28 Consider an orthogonal transformation L from R n to R n . Show that L preserves the dot product: ~v · ~w = L ( ~v ) · L ( ~w ) , for all ~v and ~w in R n . Solution: The idea is to express the dot product in terms of lengths of vectors. For any ~x and ~ y in R n , we have || ~x + ~ y || 2 = ( ~x + ~ y ) · ( ~x + ~ y ) = || ~x || 2 + 2 ( ~x · ~ y ) + || ~ y || 2 . We may write the dot product as ~x · ~ y = 1 2 || ~x + ~ y || 2 - || ~x || 2 - || ~ y || 2 . Let ~x = L ( ~v ) and ~ y = L ( ~w ). Then we have L ( ~v ) · L ( ~w ) = 1 2 || L ( ~v ) + L ( ~w ) || 2 - || L ( ~v ) || 2 - || L ( ~w ) || 2 = 1 2 || L ( ~v + ~w ) || 2 - || L ( ~v ) || 2 - || L ( ~w ) || 2 ( L is linear) = 1 2 || ~v + ~w || 2 - || ~v || 2 - || ~w || 2 ( L preserves lengths) = ~v · ~w. Problem 2. pg. 218; prob. 29 Show that an orthogonal transformation L from R n to R n preserves angles: The angle between two nonzero vectors ~v and ~w in R n equals the angle between L ( ~v ) and L ( ~w ). Conversely, is any linear transformation that preserves angles orthogonal? Solution: Let θ be the angle between ~v and ~w , and ϑ be the angle between L ( ~v ) and L ( ~w ). We want to show ϑ = θ . According to Definition 5.1.12 on page 196 of the text, we have cos θ = ~v · ~w || ~v || || ~w || and cos ϑ = L ( ~v ) · L ( ~w ) || L ( ~v ) || || L ( ~w ) || . Since L is an orthogonal transformation, we know that || L ( ~v ) || = || ~v || and || L ( ~w ) || = || ~w || . We showed in Exercise 5.3.28 (i.e., Problem 1 above) that L ( ~v ) · L ( ~w ) = ~v · ~w . This gives cos ϑ = cos θ , so ϑ = θ as desired. Conversely, a linear transformation that preserves angles is not orthogonal. As an example, con- sider the linear transformation L : R 2 R 2 defined by L ( ~x ) = 2 ~x . Then || L ( ~x ) || = 2 || ~x || so that L is not orthogonal, but L ( ~v ) · L ( ~w ) || L ( ~v ) || || L ( ~w ) || = 2 ~v · 2 ~w 2 || ~v || · 2 || ~w || = ~v · ~w || ~v || || ~w || 1

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so that L does indeed preserve angles. Remark: A linear transformation L that preserves the dot product preserves both angles and lengths – and hence must be orthogonal. Here’s why. Let ~u i = L ( ~e i ) be the image of the standard basis vector for i = 1 , 2 , . . . , n , so that the matrix of L is A = ~u 1 ~u 2 · · · ~u n . Then ~u i · ~u j = L ( ~e i ) · L ( ~e j ) = ~e i · ~e j = ( 1 if i = j , 0 otherwise. Hence the columns of A form an orthonormal basis of R n , so that L is indeed an orthogonal transformation. Combining this with Exercise 5.3.28 (i.e., Problem #1 above) we conclude that a linear transformation L : R n R n is orthogonal if and only if L preserves the dot product. Problem 3. pg. 218; prob. 31 Are the rows of an orthogonal matrix A necessarily orthonormal?
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