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# tut03 - 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 Tutorial 3 1. Determine whether the following sets are linearly independent, and whether or not they span R3 . a) X = b) Y = 1 0 0 1 2 3 , , 0 1 0 4 5 6 , , 0 0 1 1 2 -3 1 4 1 1 2 5 1 0 3 6 1 -1 2. Find the column space of A = . Do the columns of A form a basis for R3 ? 3. Let Y = {p1 (x), p2 (x), p3 (x)}, where p1 (x) = 1, p2 (x) = 2x - 1 and p3 (x) = (2x - 1)2 . a) Show that Y is a basis of P2 . b) Find the unique expression for p(x) = 5 + x + x2 P2 as a linear combination of vectors in X. 4. a) Suppose that {v1 , . . . , vn } (n 2) is a linearly independent subset of a vector space V . Is it true that if r1 v1 + r2 v2 = 0, for some r1 , r2 R, then r1 = r2 = 0? b) Suppose that {u1 , u2 , u3 } is a basis for R3 and that W is a subspace of R3 . Is it always true that some subset of {u1 , u2 , u3 } is a basis for W ? 5. For which values of R is the set -1 -1 , -1 -1 , -1 -1 linearly independent in R3 ? 6. Let g1 (x) = ex , g2 (x) = e2x and g3 (x) = e3x . Is X = {g1 , g2 , g3 } a linearly independent subset of the vector space F? Math 2061: Tutorial 3 A.M. 5/1/2012 ...
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