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tut02 - THE UNIVERSITY OF SYDNEY PURE MATHEMATICS Linear...

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Unformatted text preview: THE UNIVERSITY OF SYDNEY PURE MATHEMATICS Linear Mathematics 2012 Tutorial 2 1. Prove that the set S = braceleftBigparenleftBig x y parenrightBig ∈ R 2 | y = 4 x bracerightBig is a subspace of R 2 . 2. For each of the sets of vectors X ⊂ R 3 below, explicitly describe all of the vectors in the subspace Span( X ) of R 3 . a ) X = { } . b ) X = braceleftBigparenleftBig 1 1 1 parenrightBigbracerightBig . c ) X = braceleftBigparenleftBig 1 1 1 parenrightBig , parenleftBig 2 2 2 parenrightBigbracerightBig . d ) X = braceleftBigparenleftBig 1 1 1 parenrightBig , parenleftBig 1 parenrightBigbracerightBig . e ) X = braceleftBigparenleftBig 1 1 1 parenrightBig , parenleftBig 1 parenrightBig , parenleftBig 1 1 parenrightBigbracerightBig . 3. Recall that F is the vector space of functions from R to R , with the usual operations of addition and scalar multiplication of functions. For each of the following subsets of F , write down two functions that belong to the subset, and determine whether or not the subset is a vector subspace...
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