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Unformatted text preview: MA 265 HOMEWORK ASSIGNMENT #7 SOLUTIONS #1. Page 207; Exercise 17. Which of the following subsets of the vector space M nn are subspaces? (a) The set of all n n symmetric matrices (b) The set of all n n diagonal matrices (c) The set of all n n nonsingular matrices Solution: (a) Let W be the set of all n n symmetric matrices. In order to check that this is a subspace, we verify three properties: To show that W is nonempty, consider the n n zero matrix O n . This is a symmetric matrix, so O n W . Hence W 6 = . To show that W is closed under vector addition, consider A, B W . These are both symmetric matrices, i.e., A T = A and B T = B . Since ( A + B ) T = A T + B T = A + B , we see that A + B is a symmetric matrix. Hence A + B W as well. To show that W is closed under scalar multiplication, consider c R and A W . We see that ( cA ) T = cA T = cA , so that cA is a symmetric matrix. Hence cA W . This shows that the collection of n n symmetric matrices is a subspace. (b) Let W be the set of all n n diagonal matrices. In order to check that this is a subspace, we verify three properties: To show that W is nonempty, consider the n n zero matrix O n . This is a diagonal matrix, so O n W . Hence W 6 = . To show that W is closed under vector addition, consider A, B W . These are both diagonal matri ces, i.e., A = diag( a 1 , ..., a n ) and B = diag( b 1 , ..., b n ). Since A + B = diag( a 1 + b 1 , ..., a n + b n ), we see that A + B is a diagonal matrix. Hence A + B W as well. To show that W is closed under scalar multiplication, consider c R and A W . We see that cA = diag( ca 1 , ..., ca n ), so that cA is a diagonal matrix. Hence cA W . This shows that the collection of n n diagonal matrices is a subspace. (c) Let W be the set of all n n nonsingular matrices. To show that W is nonempty, consider the n n identity matrix, I n . This is a nonsingular matrix, so I n W . Hence W 6 = . W is not closed under vector addition: Consider A, A W . Then A + ( A ) = O n is the n n zero matrix, which is singular, so A + ( A ) 6 W . W is not closed under scalar multiplication: Consider if c = 0 and A W . Then cA = O n , so cA 6 W . This shows that the collection of n n nonsingular matrices is not a subspace. #2. Page 207; Exercise 19. ( Calculus Required ) Which of the following subsets are subspaces of the vector space C ( , ) defined in Example 7 of Section 4.2? (a) All nonnegative functions 1 2 MA 265 HOMEWORK ASSIGNMENT #7 SOLUTIONS (b) All constant functions (c) All functions f such that f (0) = 0 (d) All functions f such that f (0) = 5 (e) All differentiable functions Solution: (a) Let W be the set of all nonnegative functions....
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