prac01 - 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 1 1. Find all numbers a, b and c such that a) a 1 4 7 +b 2 5 8 +c 3 6 9 = 1 -2 1 ; b) a 1 4 7 +b 2 5 8 +c 3 6 3 = 1 -2 1 . 2. Each of the following matrices is the reduced row echelon form of an augmented matrix belonging to a system of linear equations in the variables xi , (i = 1, 2, . . .). (Both systems represented here have infinitely many solutions why?) For each augmented matrix below (i) determine the number of parameters needed to solve the system and (ii) express the solution of the system in parametric form. a) 1 0 0 1 0 0 3 4 -2 -1 0 0 x1 b) 1 0 0 0 0 1 0 0 1 2 -1 -1 2 0 2 3 4 -1 x2 3. Recall that R3 = x1 , x2 , x3 R . Describe each of the following subsets of R3 x3 geometrically and determine whether or not each subset is a vector space (under the usual addition of vectors and multiplication by scalars). x1 x1 x2 x2 a) A = R3 x2 + x2 + x2 = 1 . b) B = R3 x1 = x2 + x3 . 1 2 3 x3 x3 4. The chemical equation for the combustion of petrol is of the form x1 C8 H18 + x2 O2 - x3 CO2 + x4 H2 O, for some integers x1 , x2 , x3 and x4 . Balance this equation by solving for x1 , x2 , x3 and x4 . Hint: Balance the number of atoms of carbon (C), hydrogen (H), and oxygen (O), in that order. This gives three equations in the four unknowns x1 , x2 , x3 , and x4 . Write the linear system in the form Ax = b, where x = x1 x2 x3 x4 , and solve this equation for x. 5. Let A = 1 -3 7 -1 2 6 2 0 -4 . x1 x2 x3 a) Use Gaussian elimination to find all solutions of the equation Ax = 0, where x = and 0 = 0 . 0 b) Hence show that no nonzero linear combination of the columns of A is equal to 0. c) Let b = b1 b2 b3 0 R3 and suppose that Ax = b has at least one solution. Using (a), show that the equation Ax = b has a unique solution. Math 2061: Practice session 1 A.M. 5/1/2012 ...
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