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prac02 - T HE U NIVERSITY OF S YDNEY P URE M ATHEMATICS...

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T HE U NIVERSITY OF S YDNEY P URE M ATHEMATICS Linear Mathematics 2012 Practice session 2 1. Prove that the set S = braceleftBigparenleftBig x y z parenrightBig R 3 | 4 x - y - 5 z =0 bracerightBig is a subspace of R 3 . 2. The subspace Span( X ) of R 3 is a plane for the two choices of X R 3 given below. In both cases find the equation of this plane. a ) X = braceleftBigparenleftBig 1 1 - 1 parenrightBig , parenleftBig 2 3 5 parenrightBigbracerightBig . b ) X = braceleftBigparenleftBig 1 1 - 2 parenrightBig , parenleftBig 2 3 5 parenrightBigbracerightBig . 3. Let V be a vector space. Suppose that v 1 , v 2 , v 3 are three vectors in V such that v 3 is a linear combination of v 1 and v 2 . Prove that Span( v 1 , v 2 , v 3 )=Span( v 1 , v 2 ) . 4. Let p 1 ( x )=2 , p 2 ( x )=3+ x and p 3 ( x )= x 2 +1 be polynomial functions in P 2 . a ) Is Span( { p 1 ,p 2 ,p 3 } )= P 2 ? b ) Does the set { p 2 , p 3 } span P 2 ? 5. 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 of F . a ) The set of all constant polynomials.
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