MTH2021 Linear Algebra with Applications
MTH2025 Linear Algebra (Advanced)
Problem Set 1
School of Mathematical Sciences, Monash University
These problems cover the work from Week 1, and will be discussed in support classes during Week 2. Problems marked

MTH2021 Linear Algebra with Applications
MTH2025 Linear Algebra (Advanced)
Problem Set 2
School of Mathematical Sciences, Monash University
These problems cover the work from Week 2, and will be discussed in support classes during Week 3. Problems marked

MTH2021 Linear Algebra with Applications
MTH2025 Linear Algebra (Advanced)
Problem Set 3
School of Mathematical Sciences, Monash University
These problems cover the work from Week 3, and will be discussed in support classes during Week 4. Problems marked

MTH2021 Linear Algebra with Applications
MTH2025 Linear Algebra (Advanced)
Problem Set 2
School of Mathematical Sciences, Monash University
These problems cover the work from Week 2, and will be discussed in support classes during Week 3. Problems marked

1.9 Application: Economics (A&R 1.9)
Consider an economy consisting of n industries (producing e.g.
commodities such as coal, steel, automobiles, etc), together with
one non-producing sector (e. g. the consumer market). Economies
with one or more non-prod

MTH2025 Linear Algebra (Advanced)
Quiz 1
School of Mathematical Sciences, Monash University
This is quiz is worth 2% of your total mark for this course. You have 15 minutes to complete the quiz. Write your answers on the quiz
sheet.
1. Let k 2 R, and cons

MTH2021 Linear Algebra with Applications
Quiz 1b
School of Mathematical Sciences, Monash University
This quiz is worth 2% of your total mark for this course. There are 4 questions and you have 15 minutes to complete the quiz. Write
your answers on the qui

Theorem 2.2.8. If A is an n X n matrix and E is an n X n elementary
matrix then det(EA) = det(E) det(A).
Tm oiQME/W: %o(d(/\) j 77x (-311
& 71 7-23 (<1)
: MCE) Mg) (38 TL 22661)
W 07%er 7Lto<> i (96) Cab Jewel/Oi
MaJQLmCQA are kaonieoi 3FMT(:Q~ fig/Ga
322

MTH2021 Linear Algebra with Applications
Quiz 1a
School of Mathematical Sciences, Monash University
This quiz is worth 2% of your total mark for this course. There are 4 questions and you have 15 minutes to complete the quiz. Write
your answers on the qui

Once an LU decomposition is known for a coefcient matrix A,
solving a linear system Ax = b can be achieved by performing
substitution twice:
(i) Solve Ly = b for y using forward substitution
(ii) Solve U x = y for X using back substitution
E.g. 1.6.4. Let

Denition 4.3.7. A vector space is said to be nite dimensional if
it can be spanned by nitely many vectors.
Theorem 4.3.8. Let V be a nite dimensional vector space and let
B 2 cfw_V1, V2, . . . ,Vn be any basis.
(a) If a subset S of V has more than n vecto

Denition 3.3.3. Let A be an m x n matrix. The transformation
TA : R > Rm dened by TA(X) = Ax for all X E R" is a
matrix transformation. We call A the standard matrix for the
transformation TA.
Theorem 3.3.4. Each matrix transformation has a unique standar

E.g. 1.5.10. For each of the following matrices nd its inverse or
show that it is singular:
113 124
020, (b) 3:310.
144 558
3O Theorem 1.5.11. If a square matrix A has either a left inverse B
(so BA 2 I) or a right inverse B (so AB = I ) then A is

MTH2021 Linear Algebra with Applications
MTH2025 Linear Algebra (Advanced)
Problem Set 1
School of Mathematical Sciences, Monash University
These problems cover the work from Week 1, and will be discussed in support classes during Week 2. Problems marked

MTH2021 Linear Algebra with Applications
MTH2025 Linear Algebra (Advanced)
Assignment 1
School of Mathematical Sciences, Monash University
This assignment is worth 5% of your total mark for this course. Your completed assignment must be handed to your dem

MTH2021 Linear Algebra with Applications
MTH2025 Linear Algebra (Advanced)
Problem Set 2
School of Mathematical Sciences, Monash University
These problems cover the work from Week 2, and will be discussed in support classes during Week 3. Problems marked

MONASH UNIVERSITY SCHOOL OF MATHEMATICAL SCIENCES
2006
MTHZOIO MULTIVARIABLE CALCULUS
ASSIGNMENT 1 - 2009 [1]
Due: Wednesday 8.4.2009, 5pm
(Please return to your tutor s assignment box
in the 3' oor, mathematics building, with your
tutors name and tutoria

How to Write Mathematics
c Kevin Houston
University of Leeds
September 22, 2009
CHAPTER
0
Preface
Question: How many months have 28 days?
Mathematicians answer: All of them.
The concept
This booklet is about writing mathematics at university. At pre-unive

School of Mathematical Sciences
MTH2010 Multivariable Calculus
2017 [1]
Assignment 1
Handed out in support class the week of 13 March
To be submitted in support class the week of 27 March
The completed assignment should be submitted to your MTH2010 tutor

Denition W.1.4.7. Let G and H be groups. A function h : G > H
is a homomorphism if h(o - 7') = h(o) - h(T) for all 0,7 E G.
A bijective homomorphism is an isomorphism. Two groups are
isomorphic if there exists an isomorphism between them.
Theorem W.1.4.8.

Proof of Theorem W.2.] .1 .
30 W WA 7% Maa/W (z
\/\O(C>IA 1&5 SCH/VLQ V\>/Z~ M A66 QA+yX(V\-Fp
x); 2: saw aw/.
O'k\ L-z
Z+L Z < 64> (1%
i S P .
J=f reswgm J acmmhc
w Wt! D
H
: g
3% 66:4 3W) 54: Wale So 711M (7:)
Kw: :W Zg [email protected])ML W
BLNL imv ($6591; J ("9

W.1 Permutations
W.1.1 Functions
A function f :2 > C from a set D to a set C is a rule that assigns
to each x E D a unique element f (3:) E C. The set D is the
domain, the set C is the codomain, and f (:10) is the image of :1:
uer f. The range of f is the

MTH2021 Linear Algebra with Applications
MTH2025 Linear Algebra (Advanced)
Problem Set 1
School of Mathematical Sciences, Monash University
These problems cover the work from Week 1, and will be discussed in support classes during Week 2. Problems marked

:(u (2
Mmoloiow W 714/? K(?'
EEOQW W Com 0<\:,30<,
@g/x LLLQ/le cfw_50 $4 (4 ~ (00
g mag imam: (of<
71 MW Was Q3 (MOLUQ/wi
Corollary W. 1. 6. 4. Every 0 E St; can be written as a composition
0 w adjacent transpositions.
'42/6 onv(o):; 71% ("G/5f; 3
12 T

MTH2021 Linear Algebra with Applications
MTH2025 Linear Algebra (Advanced)
Problem Set 3
School of Mathematical Sciences, Monash University
These problems cover the work from Week 3, and will be discussed in support classes during Week 4. Problems marked

Theorem 4.2.7. A set S with two or more vectors is
(a) Linearly independent ijj no vector in S is expressible as a
linear combination of the other vectors in S.
(b Linearly dependent i at least one of the vectors in S can be
expressed as a linear combinat

4 Vector spaces
4.1 Vector spaces and subspaces (A&R 4.1, 4.2)
Denition 4.1.1 (Vector Space). LetWt equipped
with two operations named addition and scalar multi lication.
Addition is a rule that assigns to any two elements u, V E V an
element u l V. Scala

1 Linear systems and matrices
1.1 Systems of linear equations (A&R 1.1)
The original motivation for studying linear algebra was to solve
systems of simultaneous linear equations. E. g.
2x+y+z=5
4m+y+3z=9 . (1.2)
2:I:+2y+z=8
Linear equations have natural g