tut04 - THE UNIVERSITY OF SYDNEY PURE MATHEMATICS Linear...

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Unformatted text preview: THE UNIVERSITY OF SYDNEY PURE MATHEMATICS Linear Mathematics 2012 Tutorial 4 1. Let X = braceleftBigparenleftBig 6 3 2 parenrightBig , parenleftBig 2 3 1 parenrightBig , parenleftBig 4- 6 parenrightBigbracerightBig . Show that X is a linearly dependent subset of R 3 and find the dimension of Span( X ) . 2. Find a basis of the subspace V = braceleftBigparenleftBig x y z parenrightBig ∈ R 3 vextendsingle vextendsingle vextendsingle 2 x + y = 3 z bracerightBig of R 3 . 3. a ) Let X = braceleftBig parenleftbigg 1 parenrightbigg , parenleftbigg 2 1 parenrightbigg , parenleftbigg 3 2 1 parenrightbigg bracerightBig . Show that X is a linearly independent subset of R 4 . Does Span( X ) = R 4 ? What is dim Span( X )? 4. Let Y = { p , p 1 , p 2 , p 3 } , where p ( x ) = x 2- 1 , p 1 ( x ) = x , p 2 ( x ) = 3 x +2 and p 3 ( x ) = x 2 + x . a ) Show that Y spans P 2 . b ) Explain why Y is not a basis for P 2 , but that some subset of Y is a basis....
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