prac03s - THE UNIVERSITY OF SYDNEY PURE MATHEMATICS Linear...

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Unformatted text preview: THE UNIVERSITY OF SYDNEY PURE MATHEMATICS Linear Mathematics 2012 Practice session 3 — Solutions 1. Find the null space and the column space of the matrix A = parenleftBig 1 2- 2 0 3 4 3 0 6 parenrightBig . Solution parenleftBig 1 2- 2 0 3 4 3 0 6 parenrightBig −→ parenleftBig 1 2- 2 0 1 4 / 3- 6 12 parenrightBig −→ parenleftBig 1 2- 2 0 1 4 / 3 0 0 1 parenrightBig Hence, the only solution to A x = is the trivial solution, and Null( A ) = braceleftBigparenleftBig parenrightBigbracerightBig . Now suppose that, given any parenleftBig x y z parenrightBig ∈ R 3 we look for scalars a , b , c such that parenleftBig x y z parenrightBig = A parenleftBig a b c parenrightBig . We will have a unique solution for all parenleftBig x y z parenrightBig ∈ R 3 , and so Col( A ) = R 3 . 2. Find the null space and the column space of the matrix B = parenleftBig 1 2- 3 2 4- 6 3 6- 9 parenrightBig . Solution Notice that the three columns of A are multiples of one another, and so Col( B ) = braceleftBig t parenleftBig 1 2 3 parenrightBig | t ∈ R bracerightBig . (A line in R 3 .) B reduces to parenleftBig 1 2- 3 0 0 0 0 0 0 parenrightBig , and so the null space has two parameters. Null( B ) = braceleftBig s parenleftBig- 2 1 parenrightBig + t parenleftBig 3 1 parenrightBig | s, t ∈ R bracerightBig . (A plane in R 3 ). 3. In each of the following, determine whether or not X is a basis of R 3 . a ) X = braceleftBigparenleftBig 3 parenrightBig , parenleftBig 1 2 parenrightBig , parenleftBig 4 1 2 parenrightBigbracerightBig . b ) X = braceleftBigparenleftBig 1 3 parenrightBig , parenleftBig 2 1- 1 parenrightBig , parenleftBig 1- 5 4 parenrightBigbracerightBig . c ) X = braceleftBigparenleftBig 1 3 parenrightBig , parenleftBig 2 1- 1 parenrightBig , parenleftBig- 1- 2 11 parenrightBigbracerightBig . d ) X = braceleftBigparenleftBig 1 3 parenrightBig , parenleftBig 2 1- 1 parenrightBig , parenleftBig 1- 5 4 parenrightBig , parenleftBig 1 2 3 parenrightBigbracerightBig ....
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This note was uploaded on 02/06/2012 for the course MATH 2061 taught by Professor Notsure during the Three '09 term at University of Sydney.

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prac03s - THE UNIVERSITY OF SYDNEY PURE MATHEMATICS Linear...

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