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**Unformatted text preview: **A T is invertible and ( A T )-1 = ( A-1 ) T . 4. Let A = ± i 1-i ² and B = ± 1 2-i 2 i 1 + 3 i ² , 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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