MATH1081 Discrete Mathematics
2.1: Title
MATH1081
DISCRETE MATHEMATICS
Setion 2
Integers, modular arithmeti
and relations
MATH1081 Discrete Mathematics
2.2: Introduction
Number theory began as the stu
University of New South Wales School of Mathematics
MATH2601 Higher Linear Algebra
7. MATRIX EXPONENTIALS
Problem, from the firstyear algebra book. A radioactive isotope A
decays at a rate of 2% per c
UNSW Sydney School of Mathematics and Statistics
MATH2601 HIGHER LINEAR ALGEBRA
TEST 2
Students surname
Semester 1, 2017
Version A
Given name or initials
Student number
Mark
Time allowed: 40 minutes.
University of New South Wales School of Mathematics
MATH2601
Higher Linear Algebra
2. VECTOR SPACES AND LINEAR TRANSFORMATIONS
Definition. Let F be a field. A vector space over F consists of a set
V ,
University of New South Wales School of Mathematics
MATH2601 Higher Linear Algebra
5. EIGENVALUES AND EIGENVECTORS
In first year you learned about eigenvalues and eigenvectors of matrices. This year w
UNSW Sydney School of Mathematics and Statistics
MATH2601 Semester 1, 2017
Solutions to test 1
Version A
1. (a) FALSE: we have
and
(z1 z2 ) z3 = z1 z 2 z 3
z1 (z2 z3 ) = z1 z 2 z3 ,
and these are in g
4. Using the standard inner product in R2 , we have
UNSW Sydney School of Mathematics and Statistics
MATH2601 Semester 1, 2017
hT (p) | vi = p(1)v1 + p(3)v2 = (p0 + p1 )v1 + (p0 + 3p1 )v2 .
Solutions
University of New South Wales School of Mathematics
MATH2601 Higher Linear Algebra
We notice that the remainder vector is an eigenvector of A ! Writing
v2 = 3
1 , we have
Av1 = 5v1 + v2 ,
6. THE JORDA
University of New South Wales School of Mathematics
MATH2601 Higher Linear Algebra
Proof. S is a subset of R3 , which we already know is a vector
space.
REVISION OF FIRST YEAR LINEAR ALGEBRA
S contain
University of New South Wales School of Mathematics
MATH2601 Higher Linear Algebra
We notice that the remainder vector is an eigenvector of A ! Writing
v2 = 3
1 , we have
Av1 = 5v1 + v2 ,
6. THE JORDA
University of New South Wales School of Mathematics
MATH2601 Higher Linear Algebra
4. DETERMINANTS
You met determinants, and learned and used some of their important properties, in first year. In this
UNSW Sydney School of Mathematics and Statistics
MATH2601 HIGHER LINEAR ALGEBRA
TEST 1
Students surname
Semester 1, 2017
Version A
Given name or initials
Student number
Mark
Time allowed: 40 minutes.
University of New South Wales School of Mathematics
MATH2601
Therefore
W = spancfw_ (2, 1, 2) .
Higher Linear Algebra
3. INNER PRODUCTS AND ADJOINTS
Preliminary revision exercise. Using a diagram to
UNSW Sydney School of Mathematics and Statistics
MATH2601 Higher Linear Algebra
Problems
Chapter 2: Vector spaces and linear transformations
1. For each of the following, say whether the statement is
UNSW Sydney School of Mathematics and Statistics
MATH2601 Higher Linear Algebra
Answers to some problems
Chapter 1: Groups, fields and abstract algebra
1. (a) Group, identity is 0, inverse of x is x.
UNSW Sydney School of Mathematics and Statistics
MATH2601 Higher Linear Algebra
Problems
Chapter 6: The Jordan Canonical Form
1. For each of the following, show that the matrix is not diagonalisable;
UNSW Sydney School of Mathematics and Statistics
MATH2601 Higher Linear Algebra
Problems
Chapter 3: Inner products and adjoints
1. Find a basis for the orthogonal complement W of
(a) W = spancfw_ (2,
UNSW Sydney School of Mathematics and Statistics
MATH2601 Higher Linear Algebra
Problems
Chapter 4: Determinants
1. In case you need the practice, find the determinants of some of the following matric
UNSW Sydney School of Mathematics and Statistics
MATH2601 Higher Linear Algebra
Answers to some problems
Chapter 5: Eigenvalues and eigenvectors
1. For A we have
= 4, v =
3
2
and
= 5, v =
2
1
and
UNSW Sydney School of Mathematics and Statistics
MATH2601 Higher Linear Algebra
Answers to some problems
Chapter 4: Determinants
1. det A = 9, det B = 28, det C = 0, det E = 2, det F = 7.
4. x = a, x
UNSW Sydney School of Mathematics and Statistics
MATH2601 Higher Linear Algebra
Problems
Chapter 5: Eigenvalues and eigenvectors
1. (a) Find all eigenvalues and eigenvectors of the following matrices.
UNSW Sydney School of Mathematics and Statistics
MATH2601 Higher Linear Algebra
Problems
Chapter 1: Groups, fields and abstract algebra
1. Decide which of the following sets are groups under the indic
School of Mathematics and Statistics
University of New South Wales
MATH2111 Higher Several Variable Calculus
SESSION 1, 2016
Students Surname
TEST 1 Version B
Initials
Questions: 5
Pages: 2
Student Nu
Class Test 2B Solutions
Math2111 Higher Several Variable Calculus
Session 1, 2016
4.
a) Since
i
j k
a b = 1 1 2 = (0 2) i (0 + 6) j + (1 3) k = 2(i + 3 j + k),
3 1 0
the area of the parallelogram span
UNSW Sydney School of Mathematics and Statistics
MATH2601 Higher Linear Algebra
Answers to some problems
Chapter 3: Inner products and adjoints
1. (a) cfw_ (1, 8, 5) ;
(b) cfw_ (3, 1, 0, 1), (3, 1, 1,
Permuting variables & negation
Aim lecture: We define the determinant via permutations.
Fact
Let f (x1 , . . . , xn ) be an R-valued function & Sn . Then
.(f ) = ( .f ).
Proof.
Daniel Chan (UNSW)
Lec
Defn of linear maps
Prop-Defn
Let V , V 0 = F-spaces. An F-linear map or F-linear transformation is a function
T : V V 0 which satisfies any of the following equiv conditions.
1
2
3
For any F, v, v0 V
Powers of elements
Aim lecture: We introduce the notion of group actions to generalise and formalise
the notion of permuting objects such as the rows (or columns of a matrix).
Let (G , ) be a group wi
Some matrix lemmas
Aim lecture: We introduce the dimension which is an important geometric
invariant of a vector space that helps us understand the theory of linear algebra.
The basic idea is simple e