ass01 - r is any scalar and is the zero vector. (Due in the...

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The University of Sydney Summer School MATH2061: Linear Mathematics 2012 Assignment This assignment consists of Fve questions. ±ull marks will only be awarded where working is shown. One question is due in each of your Frst Fve tutorials. Attach an assignment cover sheet to your solution and hand in during your tutorial on the dates stated at the end of each question. 1. Let A = 1 0 2 0 3 - 3 4 2 7 , B = 1 0 2 0 3 - 3 4 2 6 , x = x y z , c = 0 9 8 and 0 = 0 0 0 . Solve the linear systems defned by (a) A x = c (b) A x = 0 (c) B x = c (d) B x = 0 . (Due in the tutorial Monday 9 January) 2. Prove that T = { f F | f ( x ) = ax + be x , a, b R } is a subspace oF F . (Due in the tutorial Wednesday 11 January) 3. (a) ±ind an example which shows that S = b A = B a b c d ± M 2 , 2 v v v det( A ) = ad - bc = 0 ² is not closed under addition. (b) Prove that r 0 = 0 in any vector space where
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Unformatted text preview: r is any scalar and is the zero vector. (Due in the tutorial Monday 16 January) 4. Let A = 1 0 1 2 1 1 4 3 1 . (a) Find cartesian equations of Null ( A ) and Col ( A ). (b) Are the columns of A linearly independent? Explain your answer using the denition of linear independence. (Due in the tutorial Wednesday 18 January) 5. Let X = { f, g, h } where f ( x ) = 1 + 4 x , g ( x ) = 2 + 9 x and h ( x ) = 3 + 5 x . (a) Without row reducing explain why X is linearly dependent. (b) Express h as a linear combination of f and g . (c) Find a subset of X which is a basis of P 1 . (Due in the tutorial Monday 23 January)...
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ass01 - r is any scalar and is the zero vector. (Due in the...

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