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程序代写案例-CC0678A

时间：2021-02-19

CC0678A Semester 2 2015

The University of Sydney

School of Mathematics and Statistics

MATH1014

Introduction to Linear Algebra

November 2015 Lecturers: J Parkinson, B Roberts

Time Allowed: One and a half hours

Family Name: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Other Names: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

SID: . . . . . . . . . . . . . Seat Number: . . . . . . . . . . . . .

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 are of equal value;

all questions may be attempted;

working must be shown.

Approved non-programmable, non-graphics calculators may be used.

THE QUESTION PAPER MUST NOT BE REMOVED FROM THE

EXAMINATION ROOM.

Marker’s use

only

Page 1 of 25

CC0678A Semester 2 2015 Page 12 of 25

Extended Answer Section

There are three questions in this section.

Write your answers in the spaces provided below the questions.

1. (a) Let A = (6,−1, 1), B = (3, 2, 1) and C = (4, 0,−1).

(i) Find expressions for the vectors

−→

AB,

−→

AC and

−−→

BC.

(ii) Show that ∆ABC is a right angled triangle.

CC0678A Semester 2 2015 Page 13 of 25

(b) The line L1 passes through the point (7,−3,−4) and has parametric scalar equations

x = 7 + 3s

L1 : y = −3 + s (s ∈ R).

z = −4 − s

The line L2 passes through the point (7,−3,−4) and has parametric scalar equations

x = 7 − t

L2 : x = −3 + t (t ∈ R).

z = −4 + 2t

(i) Write down vectors v1 and v2 in the direction of L1 and L2 respectively.

(ii) Find the Cartesian equation of the plane that contains the lines L1 and L2.

CC0678A Semester 2 2015 Page 14 of 25

(c) (i) Evaluate 4365 in Z7.

(ii) Solve the equation 3x+ 4 = 1 in Z9.

CC0678A Semester 2 2015 Page 15 of 25

2. (a) (i) Write down the augmented matrix for the following system of equations, and

reduce the matrix to row echelon form.

x + y − 2z = 5

x + 2y + z = 4

−x + y + 8z = −7

(ii) Write down the general solution of the above system of equations.

CC0678A Semester 2 2015 Page 16 of 25

(b) Let A be the matrix

1 −1 02 −1 1

1 −1 1

.

(i) Find the inverse of A.

(ii) Hence (or otherwise) solve the following system of linear equations.

x − y = 1

2x − y + z = 1

x − y + z = −2

CC0678A Semester 2 2015 Page 17 of 25

CC0678A Semester 2 2015 Page 18 of 25

(c) Let B =

[

−5 4

−4 3

]

and C =

[

2 1

3 2

]

.

(i) Calculate BC and CB.

(ii) Find B−1.

(iii) Suppose X is a 2 × 2 matrix such that XB = BC. Does this imply that

X = C?

Justify your answer.

CC0678A Semester 2 2015 Page 19 of 25

CC0678A Semester 2 2015 Page 20 of 25

3. (a) The females of a colony of beetles live for a maximum of 2 years. In her first year,

each female produces an average of 0.5 female offspring. In her second year, each

female produces an average of 1 female offspring. Half of the females survive their

first year to breed in the second year.

(i) Write down the Leslie matrix for the female beetle population.

(ii) Calculate the eigenvalues of L. For each eigenvalue, find a corresponding

eigenvector.

(iii) Using your answer to the previous part, show that there is a steady state

population for this colony of beetles.

(iv) What are the relative proportions of females in their first and second years in

this stable population?

CC0678A Semester 2 2015 Page 21 of 25

CC0678A Semester 2 2015 Page 22 of 25

(b) Each day for exercise I either ride my bike, swim, or run. I prefer to leave the decision

of my particular form of exercise up to chance, and so I organise my exercise schedule

as follows:

If I ride my bike on a given day then:

with probability 0.9 I will ride my bike again the next day,

with probability 0.1 I will swim the next day, and

with probability 0.0 I will run the next day.

If I swim on a given day then:

with probability 0.5 I will ride my bike the next day,

with probability 0.3 I will swim again the next day,

with probability 0.2 I will run the next day.

If I run on a given day then:

with probability 0.1 I will ride my bike the next day,

with probability 0.8 I will swim the next day,

with probability 0.1 I will run again the next day.

(i) Write down the transition matrix for the Markov chain that models my exer-

cise habits.

(ii) If I run on Monday, what is the probability that I will swim on Wednesday of

the same week?

(iii) Find the steady state probability vector for this Markov chain.

(iv) In the long run, approximately how many days per year will I ride my bike?

(Assume that a year has 365 days).

CC0678A Semester 2 2015 Page 23 of 25

CC0678A Semester 2 2015 Page 24 of 25

CC0678A Semester 2 2015 Page 25 of 25

End of Extended Answer Section

This is the last page of the question paper.

学霸联盟

The University of Sydney

School of Mathematics and Statistics

MATH1014

Introduction to Linear Algebra

November 2015 Lecturers: J Parkinson, B Roberts

Time Allowed: One and a half hours

Family Name: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Other Names: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

SID: . . . . . . . . . . . . . Seat Number: . . . . . . . . . . . . .

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 are of equal value;

all questions may be attempted;

working must be shown.

Approved non-programmable, non-graphics calculators may be used.

THE QUESTION PAPER MUST NOT BE REMOVED FROM THE

EXAMINATION ROOM.

Marker’s use

only

Page 1 of 25

CC0678A Semester 2 2015 Page 12 of 25

Extended Answer Section

There are three questions in this section.

Write your answers in the spaces provided below the questions.

1. (a) Let A = (6,−1, 1), B = (3, 2, 1) and C = (4, 0,−1).

(i) Find expressions for the vectors

−→

AB,

−→

AC and

−−→

BC.

(ii) Show that ∆ABC is a right angled triangle.

CC0678A Semester 2 2015 Page 13 of 25

(b) The line L1 passes through the point (7,−3,−4) and has parametric scalar equations

x = 7 + 3s

L1 : y = −3 + s (s ∈ R).

z = −4 − s

The line L2 passes through the point (7,−3,−4) and has parametric scalar equations

x = 7 − t

L2 : x = −3 + t (t ∈ R).

z = −4 + 2t

(i) Write down vectors v1 and v2 in the direction of L1 and L2 respectively.

(ii) Find the Cartesian equation of the plane that contains the lines L1 and L2.

CC0678A Semester 2 2015 Page 14 of 25

(c) (i) Evaluate 4365 in Z7.

(ii) Solve the equation 3x+ 4 = 1 in Z9.

CC0678A Semester 2 2015 Page 15 of 25

2. (a) (i) Write down the augmented matrix for the following system of equations, and

reduce the matrix to row echelon form.

x + y − 2z = 5

x + 2y + z = 4

−x + y + 8z = −7

(ii) Write down the general solution of the above system of equations.

CC0678A Semester 2 2015 Page 16 of 25

(b) Let A be the matrix

1 −1 02 −1 1

1 −1 1

.

(i) Find the inverse of A.

(ii) Hence (or otherwise) solve the following system of linear equations.

x − y = 1

2x − y + z = 1

x − y + z = −2

CC0678A Semester 2 2015 Page 17 of 25

CC0678A Semester 2 2015 Page 18 of 25

(c) Let B =

[

−5 4

−4 3

]

and C =

[

2 1

3 2

]

.

(i) Calculate BC and CB.

(ii) Find B−1.

(iii) Suppose X is a 2 × 2 matrix such that XB = BC. Does this imply that

X = C?

Justify your answer.

CC0678A Semester 2 2015 Page 19 of 25

CC0678A Semester 2 2015 Page 20 of 25

3. (a) The females of a colony of beetles live for a maximum of 2 years. In her first year,

each female produces an average of 0.5 female offspring. In her second year, each

female produces an average of 1 female offspring. Half of the females survive their

first year to breed in the second year.

(i) Write down the Leslie matrix for the female beetle population.

(ii) Calculate the eigenvalues of L. For each eigenvalue, find a corresponding

eigenvector.

(iii) Using your answer to the previous part, show that there is a steady state

population for this colony of beetles.

(iv) What are the relative proportions of females in their first and second years in

this stable population?

CC0678A Semester 2 2015 Page 21 of 25

CC0678A Semester 2 2015 Page 22 of 25

(b) Each day for exercise I either ride my bike, swim, or run. I prefer to leave the decision

of my particular form of exercise up to chance, and so I organise my exercise schedule

as follows:

If I ride my bike on a given day then:

with probability 0.9 I will ride my bike again the next day,

with probability 0.1 I will swim the next day, and

with probability 0.0 I will run the next day.

If I swim on a given day then:

with probability 0.5 I will ride my bike the next day,

with probability 0.3 I will swim again the next day,

with probability 0.2 I will run the next day.

If I run on a given day then:

with probability 0.1 I will ride my bike the next day,

with probability 0.8 I will swim the next day,

with probability 0.1 I will run again the next day.

(i) Write down the transition matrix for the Markov chain that models my exer-

cise habits.

(ii) If I run on Monday, what is the probability that I will swim on Wednesday of

the same week?

(iii) Find the steady state probability vector for this Markov chain.

(iv) In the long run, approximately how many days per year will I ride my bike?

(Assume that a year has 365 days).

CC0678A Semester 2 2015 Page 23 of 25

CC0678A Semester 2 2015 Page 24 of 25

CC0678A Semester 2 2015 Page 25 of 25

End of Extended Answer Section

This is the last page of the question paper.

学霸联盟