Student
Number
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Semester 1 Assessment, 2020
School of Mathematics and Statistics
MAST10022 Linear Algebra: Advanced
This exam consists of 20 pages (including this page)
Authorised materials:
• printed single-sided copy of the Exam or the Masked Exam made available earlier, or an
offline electronic pdf reader,
• 2 double-sided A4 sheets of notes (handwritten or printed), and
• blank A4 paper.
Instructions to Students
• During exam writing time you may only interact with the device running the Zoom session
with supervisor permission. The screen of any other device must be visible in Zoom from
the start of the session.
• If you have a printer, print out the exam single-sided and hand write your solutions into
the answer spaces.
• If you do not have a printer, or if your printer fails on the day of the exam,
(a) download the exam paper to a second device (not running Zoom), disconnect it from
the internet as soon as the paper is downloaded and read the paper on the second device;
(b) write your answers on the Masked Exam PDF if you were able to print it single-sided
before the exam day.
If you do not have the Masked Exam PDF, write single-sided on blank sheets of paper.
• If you are unable to answer the whole question in the answer space provided then you can
append additional handwritten solutions to the end of your exam submission. If you do
this you MUST make a note in the correct answer space or page for the question, warning
the marker that you have appended additional remarks at the end.
• Assemble all the exam pages (or template pages) in correct page number order and the
correct way up, and add any extra pages with additional working at the end.
• Scan your exam submission to a single PDF file with a mobile phone or a scanner. Scan
from directly above to avoid any excessive keystone effect. Check that all pages are clearly
readable and cropped to the A4 borders of the original page. Poorly scanned submissions
may be impossible to mark.
• Upload the PDF file via the Canvas Assignments menu and submit the PDF to the
GradeScope tool by first selecting your PDF file and then clicking on Upload PDF.
• Confirm with your Zoom supervisor that you have GradeScope confirmation of submission
before leaving Zoom supervision.
• You should attempt all questions.
• Marks may be awarded for: using appropriate mathematical techniques, accuracy of the
solution, showing full working, including results used, and correct use of mathematical
notation.
• There are 12 questions with marks as shown. The total number of marks available is 100.
Supplied by download for enrolled students only— c University of Melbourne 2020
MAST10022 Linear Algebra: Advanced
Semester 1, 2020
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Question 1 (6 marks)
Consider the following system of linear equations in the unknowns x, y, z ∈ R (with an unspecified parameter k ∈ R):
x − y = k − 11
x+z =8
y + z = 13
(a) Find the values of k for which the system is (i) consistent; (ii) inconsistent.
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MAST10022 Linear Algebra: Advanced
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(b) In case (i) find the general solution.
Question 2 (7 marks)



1
2 −1 1
1
Let M = 1 2 −1 and N = 
1
0 1
3
a

2 −1 1
1 2 −1
 where a ∈ R.
0 1
3
0 0
0
(a) Use row operations to calculate det(M ).
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Semester 1, 2020
MAST10022 Linear Algebra: Advanced
Semester 1, 2020
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(b) Write det(N ) in terms of det(M ) and a.
Question 3 (10 marks)
In each part of this question, determine whether or not W is a subspace of the 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 | 20ab = 21cd}
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MAST10022 Linear Algebra: Advanced
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(b) V = M3,3 (C), W = {A ∈ M3,3 (C) | A2 = 3A}
(c) V = P3 (R), W = {p ∈ P3 (R) | p(20) = p(19)}
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Semester 1, 2020
MAST10022 Linear Algebra: Advanced
Semester 1, 2020
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Question 4 (7 marks)
You are given that the following matrices are related by a sequence of elementary row operations:




1 0 −2 0
1 0 −2 2
0 1 3 0
−2 1 7
0


B=
A=
0 0 0 1
 1 2 4 11
0 0 0 0
1 1 1
6
Using this information (or otherwise), answer the following questions.
(a) Show that the set
1 − 2x + x2 + x3 , x + 2x2 + x3 , −2 + 7x + 4x2 + x3 , 2 + 11x2 + 6x3
is not a linearly independent subset of P3 (R) by writing one of the elements as a linear
combination of the others.
(b) Is the set
{(1, −2, 1, 1), (0, 1, 2, 1), (−2, 7, 4, 1), (2, 0, 11, 6)}
a spanning set for R4 ? (Be sure to justify your answer.)
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MAST10022 Linear Algebra: Advanced
Semester 1, 2020
(c) Find a basis for the subspace of M2,2 (R) defined by
1 0
−2 1
1 2
1 1
W = Span
,
,
,
−2 2
7 0
4 11
1 6
What is the dimension of W ?
Question 5 (6 marks)
Consider the matrices H ∈ M3,6 (F2 ) and u, v ∈ M6,1 (F2 ) given by


1 0 0 0 1 1
H= 0 1 0 1 0 1 
0 0 1 1 1 0
 
0
0
 
1

u=
1
 
1
1
where F2 = {0, 1} is the field with two elements.
(a) Calculate Hu and Hv.
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 
1
0
 
1

v=
1
 
0
1
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MAST10022 Linear Algebra: Advanced
Semester 1, 2020
(b) Let C be the binary linear code that has H as its check matrix. For each of the following
words, decide if the word is a codeword. If it is not a codeword, find the nearest
codeword. (You should explain your answers.)
(i) 001111
(ii) 101101
(c) How many codewords does the code C contain?
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MAST10022 Linear Algebra: Advanced
Semester 1, 2020
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Question 6 (10 marks)
Consider the function T : P2 (R) → P2 (R) given by
T (p) =
d2 p
dp
+x
−p
2
dx
dx
Let B = {1, x, x2 } be the standard basis of P2 (R).
(a) Prove that T is a linear transformation.
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MAST10022 Linear Algebra: Advanced
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(b) Find the matrix [T ]B of T with respect to the basis B.
(c) Find a basis for the image of T .
(d) Find a basis for the kernel of T .
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MAST10022 Linear Algebra: Advanced
Semester 1, 2020
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(e) Is T injective? Is T surjective? (Be sure to explain your answers.)
Question 7 (9 marks)
Let B = {b1 , b2 , b3 } be a basis for a real vector space V , and suppose that
c1 = b1 ,
c2 = b1 − b2 ,
c3 = b1 − 3b2 + b3
(a) Show that C = {c1 , c2 , c3 } is a basis for V .
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MAST10022 Linear Algebra: Advanced
Semester 1, 2020
(b) Find the transition matrix PB,C (that converts coordinates with respect to C to coordinates with respect to B).
(c) Find the transition matrix PC,B (that converts coordinates with respect to B to coordinates with respect to C).
(d) Find the coordinate matrix of v = 2b1 + 3b2 − b3 with respect to C.
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MAST10022 Linear Algebra: Advanced
Semester 1, 2020
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Question 8 (7 marks)
Let T : P1 (R) → P1 (R) be the linear transformation given by
T (a + bx) = a + b + (2a + b)x
Consider the following two bases of P1 (R)
S = {1, x}
and B = {1 + 2x, 3 + 4x}
(a) Find the matrix [T ]S .
(b) Is T invertible? (You should explain your reasoning.)
(c) Find the matrix [T ]B .
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MAST10022 Linear Algebra: Advanced
Semester 1, 2020
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Question 9 (10 marks)
Consider P1 (R) with the inner product given by
hp, qi = p(1)q(1) + 2p(2)q(2)
(a) Prove that the formula for hp(x), q(x)i defines an inner product on P1 (R).
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MAST10022 Linear Algebra: Advanced
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(b) Calculate the length of x with respect to the inner product.
(c) Let W ⊆ P1 (R) be the subspace spanned by x.
Find the element of W that is closest to x + 1.
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MAST10022 Linear Algebra: Advanced
Question 10

1

Let A = 1
1
Semester 1, 2020
(10 marks)

1 1
1 1 ∈ M3,3 (R).
1 1
(a) Explain how we can tell, without the need for any calculation, that A is diagonalisable
over R.
(b) Find the characteristic polynomial of A.
(Note: In this and the next part, you may use that (1, 1, 1) is an eigenvector of A.)
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MAST10022 Linear Algebra: Advanced
Semester 1, 2020
(c) Find a basis B for R3 such that B is orthonormal (with respect to the dot product on
R3 ) and all elements of B are eigenvectors of A.
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MAST10022 Linear Algebra: Advanced
Semester 1, 2020
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Question 11 (8 marks)
Let A ∈ Mn,n (F) where F is a field.
(a) Prove that if λ ∈ F is an eigenvalue of A, then λ2 is an eigenvalue of A2 .
(b) Is it always true that every eigenvector of A2 is also an eigenvector of A? Justify your
answer by either giving a general proof, or by giving an example of a matrix A where
this does not hold.
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MAST10022 Linear Algebra: Advanced
Semester 1, 2020
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Question 12 (10 marks)
Let U , V be vector spaces and S : U → V a linear transformation. Suppose that V is finite
dimensional and that S is injective.
(a) Show that U is finite dimensional.
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MAST10022 Linear Algebra: Advanced
Semester 1, 2020
(b) Suppose further that T : V → W is a surjective linear transformation and that im(S) =
ker(T ). Show that
dim(V ) = dim(U ) + dim(W )
End of Exam—Total Available Marks = 100
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