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

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Unformatted text preview: T HE U NIVERSITY OF S YDNEY P URE M ATHEMATICS Linear Mathematics 2012 Practice session 2 1. Prove that the set S = x y z R3 | 4x - y - 5z = 0 is a subspace of R3 . 2. The subspace Span(X) of R3 is a plane for the two choices of X R3 given below. In both cases find the equation of this plane. 1 1 2 2 1 1 b) X = a) X = , 3 . , 3 . -2 -1 5 5 3. Let V be a vector space. Suppose that v1 , v2 , v3 are three vectors in V such that v3 is a linear combination of v1 and v2 . Prove that Span(v1 , v2 , v3 ) = Span(v1 , v2 ). 4. Let p1 (x) = 2, p2 (x) = 3 + x and p3 (x) = x2 + 1 be polynomial functions in P2 . a) Is Span({p1 , p2 , p3 }) = P2 ? b) Does the set {p2 , p3 } span P2 ? 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. b) The set of polynomials whose coefficients are all non-negative. c) The set { f : R - R | f (0) = 1 }. d) The set { f : R - R | f (-x) = f (x) for all x R }. This is the set of all even functions. 6. Let A = 1 0 1 0 3 1 2 1 -2 -2 0 -2 and write v1 , v2 , v3 for the columns of A, in the obvious order. a) Recall that the column space of A is defined to be the subspace Span({v1 , v2 , v3 }). Determine which of the following vectors belongs to the column space of A. i) v4 = ii) v5 = 3 2 1 2 0 1 0 1 4 2 3 2 b) Let Y = {v1 , v2 , v6 }, where v6 = . Show that the subspace Span(Y ) contains the column space of A. c) Show that there are vectors in R4 which do not belong to Span(Y ). Math 2061: Practice session 2 A.M. 5/1/2012 ...
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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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