Student
Number
Semester 2 Assessment, 2019
School of Mathematics and Statistics
MAST10007 Linear Algebra
Writing time: 3 hours
This is NOT an open book exam
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.
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
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