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Unformatted text preview: 8009A SEMESTER 2 2009 THE UNIVERSITY OF SYDNEY
SCHOOL OF MATHEMATICS AND STATISTICS MATH1014
INTRODUCTION To LINEAR ALGEBRA November 2009 LECTURERS: S Britten, J Henderson, N Saunders TIME ALLOWED: One and a half hours Family Name: ............................................................ Other Names: This examination has two sections: Multiple Choice and Extended Answer. The Multiple Choice Section is worth 50% of the total examination;
there are 25 questions; the questions are of equal value;
all questions may be attempted. Answers to the Multiple Choice questions must be entered on
the Multiple Choice Answer Sheet. The Extended Answer Section is worth 50% of the total examination;
there are 3 questions; the questions areiof 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 20 I MARKER’S USE 8009A SEMESTER 2 2009 PAGE 12 OF 20 Ext ended Answer Section There are three questions 2n this section.
Write your answers in the spaces provided below the questions. MARK
1. (a) (2) A = (3,1,4) and B = (2, 2, 5) are points in R3.
Find the cross product of 571) and O_B>, where 0 is the origin, (0,0,0). 2
(22) Hence ﬁnd the Cartesian equation of the plane through the origin 0 and the
points A and B. 2
(b) (2') Find the vector equation of the line joining the points P = (5, ~1, 2) and
Q=(2,7,1)mn3. 2
(22) For which value of the constant k is the line joining P and Q parallel to the
plane23—3y+hz=1? 3
(c) The lines 772 and n in R3 are given by the following vector equations:
3 1
m : X = 2 + s —3
l 2
and
—1 2
n x = O + t l ,
3 —3
where s and t can take any real value.
(2) Say why the lines 272 and n are not parallel. 1 (22) Show that the lines m and n do not intersect. 2 8009A SEMESTER 2 2009 PAGE 13 OF 20 8009A SEMESTER 2 2009 PAGE 14 OF 20 8009A SEMESTER 2 2009 PAGE 15 OF 20 MARK
2. (a) Write down the augmented matrix for the following system of equations, and reduce
the matrix to row echelon form. Hence solve the system of equations.
a: + y — z = 0
3:1: + 2y — 4z 2 4
2:1: — y + 52 = 2
4 8009A SEMESTER 2 2009 PAGE 16 OF 20 MARK
(b) The augmented matrix for a system of linear equations in as, y and z reduced to the
following row echelon form:
1 —2 1 2
O 1 3 4
O 0 O 0
Solve the system of equations. 2
2 0 O 2 —1 7 3 —1
(c)LetA= O3 8 , B: 1 0 3 , C: 1 2
0 1 1 —2 4 0 2 2
(2') Find A — 3B. 1
(2’2“) Find BC. 2
(m) Find the determinant of A. 1 (iv) Find the eigenvalues of A. 2 8009A SEMESTER 2 2009 PAGE 17 OF 20 8009A SEMESTER 2 2009 PAGE 18 OF 20 3. 3 20 O
(a) The Leslie matrix 0.5 O O has an eigenvalue of 5.
O 0.5 0 Find a corresponding eigenvector. (b) Near Xavier’s house there are two coffee shops, The Jolly Roaster and The Daily
Grind. Each day, Xavier buys a cup of coffee from one of the two shops. If he buys
from The Jolly Roaster one day, he always buys from The Daily Grind the following
day. If he buys from Tlhe Daily Grind one day, the chance that he will buy from it again the next day is Z. (i) Write down the transition matrix for the Markov chain that models Xavier’s
coffee—buying habits. (ii) If Xavier buys his coﬁee at The Daily Grind on Monday, what is the probability
that he buys his coffee at The Daily Grind on Wednesday of the same week? (iii) Find the steady state probability vector for this Markov chain. (iv) In the long run, on how many days of each week does Xavier buy his coffee
at The Jolly Roaster? (c) Let A be a stochastic matrix. (A stochastic matrix is a square matrix in which all
the entries are non—negative, and the entries in each column add to 1.) (i) Prove that the sum of the components of Ax is equal to the sum of the
components of x. (i2) Deduce that if Ax = Ax, and A 75 1, then the sum of the components of x is
zero. MARKS ...
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