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Assignment 2 - A T is invertible and A T-1 = A-1 T 4 Let A...

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MATH 223, Linear Algebra Winter 2008 Assignment 2, due in class Wednesday, January 23, 2008 C stands for the set of complex numbers, and R for the set of real numbers. 1. Here are some subsets of the complex vector space C 3 . In each case, decide whether the given set S is a subspace of C 3 ; justify your answers. (a) S = { a - b 3 ib (2 + i ) a - 2 b : a, b any complex numbers } . (b) S = { a - b 3 ib + 2 (2 + i ) a - 2 b : a, b any complex numbers } . (c) S = { a - b 3 ib (2 + i ) a - 2 b : a, b any real numbers } . (d) S = { ~v ∈ C 3 : 1 2 3 i 2 i 3 i ~v = 0 2 - 2 i 4 i 1 - 4 i 0 ~v } . 2. Let R R be the (real) vector space of all functions from the real numbers to the real numbers. For each of the following subsets of R R , decide whether it is a subspace; justify your answers. (a) The set of all even functions. (A real-valued function is called even if it satisfies f ( - x ) = f ( x ) for all x .) (b) { f ∈ R R : f (0) = 7 } . (c) { f ∈ R R : f (7) = 0 } . 3. Suppose that A is an invertible matrix over any field. Show that
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Unformatted text preview: A T is invertible and ( A T )-1 = ( A-1 ) T . 4. Let A = ± i 1-i ¶ and B = ± 1 2-i 2 i 1 + 4 i ¶ , both invertible matri-ces with complex entries. Find AB , A-1 , B-1 , ( AB )-1 , A-1 B-1 and B-1 A-1 . Express A and B and each of these matrices as a product of elementary matrices. 5. Suppose that V is any vector space over any field, and W 1 and W 2 are subspaces of V . (a) Show that the intersection W 1 ∩ W 2 is also a subspace of V . (b) Show that the union W 1 ∪ W 2 is almost never a subspace of V — specifically, show that this only occurs if W 1 ⊆ W 2 or W 2 ⊆ W 1 . 1 (c) We let the sum W 1 + W 2 be defined as the set { ~w 1 + ~w 2 : ~w 1 ∈ W 1 , ~w 2 ∈ W 2 } . Show that this set is a subspace of V . 2...
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