Student

Number

Semester 2 Assessment, 2019

School of Mathematics and Statistics

MAST10007 Linear Algebra

Writing time: 3 hours

Reading time: 15 minutes

This is NOT an open book exam

This paper consists of 6 pages (including this page)

Authorised Materials

• Mobile phones, smart watches and internet or communication devices are forbidden.

• No written or printed materials may be brought into the examination.

• No calculators of any kind may be brought into the examination.

Instructions to Students

• You must NOT remove this question paper at the conclusion of the examination.

• There are 13 questions on this exam paper.

• All questions may be attempted.

• Start each question on a new page. Clearly label each page with the number of the

question that you are attempting.

• Marks may be awarded for

– Using appropriate mathematical techniques.

– Accuracy of the solution.

– Full explanations, including justification of rules or theorems used.

– Using correct mathematical notation.

• The total number of marks available is 120.

Instructions to Invigilators

• Students must NOT remove this question paper at the conclusion of the examination.

• Initially students are to receive the exam paper and two 11 page script books.

This paper may be held in the Baillieu Library

MAST10007 Semester 2, 2019

Question 1 (9 marks)

Consider the system of equations

x + 2y + 2z = 2

2x + 5y + 3z = 5

x + 3y + k2z = k + 2

where x, y, z ∈ R and k ∈ R.

(a) Determine the values of k, if any, for which the system has

(i) a unique solution, (ii) no solutions, (iii) infinitely many solutions.

(b) Find all solutions to the system when k = 1.

Question 2 (11 marks)

(a) Consider the matrices

A =

[

1 2

3 5

]

, B =

[

2 0 −1

1 −2 4

]

, C =

[

1 2 3

0 5 −1

]

.

Calculate the following, if they exist:

(i) AB, (ii) BCT .

(b) Prove that if A and B are matrices such that AB and BA are both defined, then AB and

BA are both square matrices.

(c) Use cofactor expansion to find the determinant of the matrix

A =

a 0 b 0

0 a 0 b

c 0 d 0

0 c 0 d

,

where a, b, c, d are complex numbers.

When is A invertible? Explain your answer.

Question 3 (5 marks)

Let

A =

0 1 11 0 1

1 1 0

.

(a) Verify that A2 −A = 2I, where I is the 3× 3 identity matrix.

(b) Deduce from part (a) that A is invertible, and that A−1 = 12(A− I).

Page 2 of 6 pages

MAST10007 Semester 2, 2019

Question 4 (12 marks)

Consider the points P (1, 1, 0), Q(0, 1, 2) and R(1, 0, 1) in R3.

(a) Find the distance between P and Q.

(b) Find the cosine of the angle between the vectors

−→

PQ and

−→

PR.

(c) Find the area of the triangle with vertices P , Q and R.

(d) Find a Cartesian equation for the plane that passes through P , Q and R.

Question 5 (10 marks)

In each part of this question, determine whether W is a subspace of the real vector space V .

For each part, give a complete proof using the subspace theorem, or a specific counterexample

to show that some subspace property fails.

(a) V = R4, W = {(a, b, c, d) ∈ R4 | a+ b+ c+ d = 1}.

(b) V = P3, W = {p(x) ∈ P3 | p(x) = p(−x) for all x ∈ R}.

(c) V = M2,2, W = {A ∈M2,2 | A2 = A}.

Question 6 (9 marks)

The following two matrices are related by a sequence of elementary row operations:

A =

1 2 −1 0

2 1 4 3

1 0 3 2

2 −1 8 5

, B =

1 0 3 2

0 1 −2 −1

0 0 0 0

0 0 0 0

Let W be the subspace of R4 spanned by the set of vectors

S = {(1, 2, 1, 2), (2, 1, 0,−1), (−1, 4, 3, 8), (0, 3, 2, 5)}.

(a) Find a subset of S that is a basis for W . Hence, find the dimension of W .

(b) Write the other vectors in S as a linear combination of your basis vectors.

(c) Consider the set of vectors

T = {(−1, 4, 3, 8), (0, 3, 2, 5)}.

Is T linearly independent? Is T a basis for W? Explain your answers.

Page 3 of 6 pages

MAST10007 Semester 2, 2019

Question 7 (11 marks)

Let T : R3 → R3 be the linear transformation given by

T (x, y, z) = (x+ y + 2z, y + 2z, z).

Consider the bases of R3 given by

S = {(1, 0, 0), (0, 1, 0), (0, 0, 1)}

and

B = {(1, 0, 0), (1,−1, 0), (2,−1,−1)} .

(a) Find the matrix [T ]S of T with respect to the standard basis S.

(b) Is T (i) injective, (ii) surjective, (iii) invertible? Explain your answers.

(c) Find the transition matrix PS,B.

(d) Find the transition matrix PB,S .

(e) Find the matrix [T ]B of T with respect to the basis B.

Question 8 (13 marks)

Consider the function T : M2,2 →M2,2 given by

T (X) =

[

2 2

1 1

]

X.

(a) Show that T is a linear transformation.

(b) Find the matrix [T ]S of T with respect to the standard basis

S =

{[

1 0

0 0

]

,

[

0 1

0 0

]

,

[

0 0

1 0

]

,

[

0 0

0 1

]}

(c) Find a basis for the kernel of T .

(d) Find a basis for the image of T .

(e) Verify the rank-nullity theorem for the linear transformation T .

Question 9 (6 marks)

For each of the following matrices, determine whether it is diagonalisable and give a short

justification:

A =

[

i 1 + i

0 i

]

, B =

−2 5 10 3 −6

0 0 −4

, C =

−2 5 15 −2 −6

1 −6 −2

.

(Hint: Very little calculation should be needed to answer this question.)

Page 4 of 6 pages

MAST10007 Semester 2, 2019

Question 10 (13 marks)

In a certain town, the weather each day is either rainy or fine.

• If the weather is rainy one day, then it is rainy the next day 60% of the time.

• If the weather is fine one day, then it is fine the next day 80% of the time.

Let rn be the probability that the weather is rainy after n days, and fn be the probability that

the weather is fine after n days.

(a) Explain briefly why [

rn+1

fn+1

]

= A

[

rn

fn

]

,

where

A =

[

0.6 0.2

0.4 0.8

]

.

(b) Find the eigenvalues and corresponding eigenvectors for A.

(c) Find an invertible matrix P and a diagonal matrix D such that A = PDP−1.

(d) Assuming that today is fine we have r0 = 0 and f0 = 1. Find formulas for rn and fn for

n ≥ 1.

(e) What are the long term probabilities of rainy days rn and fine days fn, as n→∞?

Question 11 (10 marks)

Consider R3 with the standard inner product given by the dot product

〈u,v〉 = u · v = u1v1 + u2v2 + u3v3.

Let W ⊂ R3 be the subspace spanned by

{(0, 1, 1), (1, 0, 1)}.

(a) Find an orthonormal basis for W .

(b) For v = (1, 1, 0) ∈ R3, find

(i) the orthogonal projection of v onto W ,

(ii) the distance from v to W .

Page 5 of 6 pages

MAST10007 Semester 2, 2019

Question 12 (7 marks)

(a) Find the least squares line of best fit y = a+ bx for the data points

{(−1, 2), (0, 1), (1, 2), (2, 3)}

(b) Draw a clear graph showing the data points and your line of best fit.

Question 13 (4 marks)

Let A be an n× n real matrix. Fix a real number λ and consider the set

W =

{

w ∈ Rn | (A− λI)2w = 0} .

Show that W 6= {0} if and only if λ is an eigenvalue of A.

End of Exam—Total Available Marks = 120

Page 6 of 6 pages