Test_2-QA

# Test_2-QA - MATH1111/Linear Algebra/25Oct10/Test 2/a 1 THE...

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MATH1111/Linear Algebra/25Oct10/Test 2/a 1 THE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS Test 2: Suggested solution ± The set V of 2 ² 2 nonsingular matrices ± a b c d ² under the usual matrix operations is a vector space. F Explanation : Clearly V ³ R 2 ± 2 . If V is a vector space w.r.t. the usual matrix operations, then V is a subspace of R 2 ± 2 . So the zero matrix 0 2 V . The subspaces of a vector space share the same zero element. However, 0 is singular! (Compare with Tutorial II Qn 3 (b).) ± The set V 0 of 2 ² 2 matrices ± a b c d ² where b + c = 0 under the usual matrix operations is a vector space. T Explanation : You may give a proof in a way similar to Tutorial II Qn 3 (c). Alternatively, observe that it su±ces to show V 0 is subspace of R 2 ± 2 . ± Let x and y be vectors in a vector space. If x 6 = y , then Span( x ) + Span( y ) is a subspace of dimension 2. F Explanation : Consider the vector space R 2 , x = ± 1 0 ² and y = 2 x . Then Span( x ) + Span( y ) = Span( x ), which is a subspace of dimension 1.

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Test_2-QA - MATH1111/Linear Algebra/25Oct10/Test 2/a 1 THE...

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