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098026A - 8026A SEMESTER 1 2009 THE UNIVERSITY OF SYDNEY...

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Unformatted text preview: 8026A SEMESTER 1 2009 THE UNIVERSITY OF SYDNEY FACULTIES OF ARTS, ECONOMICS, EDUCATION, ENGINEERING AND SCIENCE MATH 1902 LINEAR ALGEBRA (ADVANCED) June 2009 LECTURERS: J East, A Molev TIME ALLOWED: One and a half hours This examination has two sections: Multiple Choice and Extended Answer. The Multiple Choice Section is worth 35% of the total examination; there are 20 questions; the questions are of equal value; all questions may be attempted. Answers to the Multiple Choice questions must be coded onto the Multiple Choice Answer Sheet. The Extended Answer Section is worth 65% of the total examination; there are 4 questions; the questions are of equal value; all questions may be attempted; working must be shown. Calculators will be supplied; no other calculators are permitted. THE QUESTION PAPER MUST NOT BE REMOVED FROM THE EXAMINATION ROOM. PAGE 1 OF 7 8026A SEMESTER 1 2009 PAGE 6 OF 7 Extended Answer Section Answer these questions in the answer book(s) provided. Ask for extra books if you need them. 1. (10 marks). (a) Consider the planes 791 and P2 described by the equations ('6) (ii) (iii) az+2y—z=3 and 2m—y+82=1. Find vectors 111 and 112 such that ul J. 731 and u; _L 732. Explain why 731 and 772 are not parallel. Find the parametric vector equation of the line L which is the intersection of 731 and 732. Consider the plane 733 given by the equation 3:2: — 2y — z = 5. Without explicitly calculating the intersection, explain Why the intersection of all three planes is a single point. (b) (z) Consider the points A(2, 3) and B(—2, 1) in the plane. Find the area of the parallelogram that has 0A and OB as two of its sides. (ii) Find the angle AOB of the parallelogram in the previous part. 2. (10 marks). (a) Let u and v be non—zero, perpendicular vectors in the plane. Show that if au+bv=cu+dv for scalars a, b, c, d, then a = c and b = d. (b) Let ABC'D be a square, and suppose that M and N divide AB and AD internally, and non—trivially, in the ratios a : ,6 and 'y : 6, respectively, where a+,8 = 7+6 = 1. Let P be the point of intersection of DM and EN. (2') (12') Draw a neat diagram displaying this information. Find scalars p,q,r,s with p + q = r + .3 = 1 such that P divides BM in ——-> the ratio p : q and BN in the ratio 7" : s. (Hint: write p = AP as a linear —) ——r combination of b 2 AB and d 2 AD in two ways, and apply part (a).) 8026A SEMESTER 1 2009 PAGE 7 OF 7 3. (10 marks). (a) Give the definition of a left inverse and a right inverse of a matrix A. (b) Prove that if a. matrix A has a left inverse and a right inverse then they are equal. (0) Find all values of a: for which the matrix ‘ 1 —2 A: 2 —4 —3 a: has a. left inverse. (d) Take a value of x for which the matrix A in the previous part has a left inverse. Explain why the number of left inverse matrices of A is infinite. 4. (10 marks). The matrix C’ is given by 110 02121 011 (a) Calculate the characteristic polynomial det(C' — III) of the matrix C. (b) Find the eigenvalues of C'. (G) Let k be a positive integer. Calculate the eigenvalues of the matrix C’k. Justify your calculation. (d) Hence, give a formula for the characteristic polynomial of the matrix C5. End of Extended Answer Section THIS IS THE LAST PAGE OF THE QUESTION PAPER. ...
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