1111_08_t2sol

1111_08_t2sol - MATH1111/24Oct08/Test2 1 MATH1111 Linear...

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MATH1111/24Oct08/Test2 1 MATH1111 Linear Algebra - Test 2 Solution Outline 1. (18 marks) Let x = 2 1 - 2 , y = 4 3 2 , z = 1 1 1 . (a) Are x ,y ,z linearly independent? Explain your answer. (b) Describe explicitly Span( z ,x ) Span( z ,y ). Explain your answer. Also, let u = - 1 1 10 , v = 6 4 0 . (c) Is Span( x ,y ) = Span( u ,v )? Explain your answer. Ans . (a) Yes. Consider c 1 x + c 2 y + c 3 z = 0 . It can be written as 2 4 1 1 3 1 - 2 2 1 c 1 c 2 c 3 = 0 0 0 . The matrix is singular, as its determinant = - 2 6 = 0. c 1 ,c 2 ,c 3 must be all zero. (b) Let t Span( z ,x ) Span( z ,y ). Then t = az + bu = cz + dv . Thus, ( a - c ) z + bu + dv = 0 . By part (a), a - c = 0, b = d = 0. i.e. t = az . So, Span( z ,x ) Span( z ,y ) Span( z ). Obviously, Span( z ) Span( z ,x ) Span( z ,y ). Hence, Span( z ,x ) Span( z ,y ) = Span( z ). (c) Yes. Note that
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1111_08_t2sol - MATH1111/24Oct08/Test2 1 MATH1111 Linear...

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