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Ma1bAnHw1 - S is a subspace but show your answer is correct...

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CALIFORNIA INSTITUTE OF TECHNOLOGY Department of Mathematics Math 1b, Analytical; Homework Set 1 Due: 10am, Monday, January 14, 2008 Read the first 9 sections of Chapter 1 of Apostol. That is read pages 3 through 13. You can collaborate on the problems as long as you write up all solutions in your own words and understand those solutions. Remember to justify all your answers. 1. Let U be a nonempty collection of subspaces of a vector space V . Prove W = \ U ∈U U is a subspace of V. 2. Let F be a family of m linear equations n X j =1 a i,j x j = b i , 1 i m, in variables x 1 , . . . , x n with a i,j , b i F = R or C . A solution to F is a vector v = ( v 1 , . . . , v n ) V n such that n j =1 a i,j v j = b i for all i . Determine for which ( b 1 , . . . , b m ) V m the set S ( F ) of solutions of F is a subspace of V n . Show that your answer is correct. 3. Problems 1-10 on page 13. Do not compute the dimension of
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Unformatted text preview: S is a subspace, but show your answer is correct. 4. Fix a positive integer n and define the product of two n by n matrices ( a i,j ) and ( b i,j ) by ( a i,j ) · ( b i,j ) = ( c i,j ), where c i,j = n X k =1 a i,k b k,j This product is a binary operation on M n,n called matrix multiplication and is discussed in Chapter 16, but don’t appeal to the results there. (1) Show M n,n has an identity under matrix multiplication called the identity matrix . Describe the identity matrix. (2) Show that if n > 1 then matrix multiplication is not commutative. (3) Assume n = 2, find which 2 by 2 matrices have an inverse with respect to matrix multiplication, and find the inverse under multiplication of each such matrix. 1...
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