prac03 - 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 3 1. Find the null space and the column space of the matrix A = 2. Find the null space and the column space of the matrix B = 1 2 -2 0 3 4 3 0 6 1 2 -3 2 4 -6 3 6 -9 . . 3. In each of the following, determine whether or not X is a basis of R3 . 4 1 3 0 , 2 , 1 . a) X = b) X = c) X = d) X = 0 1 0 3 1 0 3 1 0 3 , , , 0 2 1 -1 2 1 -1 2 1 -1 2 , , , 1 -5 4 -1 -2 11 1 -5 4 . . , 1 2 3 . 4. Let f1 (x) = cos x, f2 (x) = cos(x + 1) and f3 (x) = sin x. Show that {f1 , f2 , f3 } is a linearly dependent subset of F. 5. Let Y = {f1 (x), f2 (x), f3 (x)}, where f1 (x) = sin x, f2 (x) = sin 3x and f3 (x) = sin 5x. Show that Y is a linearly independent subset of F. 6. Let X = {p1 , p2 , p3 }, where p1 (x) = x2 - 1, p2 (x) = x(x - 1), p3 (x) = x(x + 1). a) Show that X is a basis of P2 . b) Find the unique expression for p(x) = 9x2 - x - 4 as a linear combination of vectors in X. Math 2061: Practice session 3 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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