MATH 2040 Linear Algebra II
Supplementary Notes by Martin Li
Polynomials
1
The notion of polynomial must already be familiar to you. In this note, we will give some properties
of polynomial that would
MATH 2040 Linear Algebra II
Lecture Notes by Martin Li
Lecture 4
Bases and dimension
1
1
Basis
In the last lecture, we have introduced the notions of linear independence and span. Now we combine
these
MATH 2040 Linear Algebra II
Lecture Notes by Martin Li
Lecture 11
Inner product spaces
1
We have generalized the vector space structure (i.e. vector addition and scalar multiplication) of
Rn to arbitr
MATH 2040 Linear Algebra II
Lecture Notes by Martin Li
Lecture 2
Subspaces 1
In the study of any algebraic structure, it is often interesting to examine subsets that possess the
same structure as the
MATH 2040 Linear Algebra II
2016-17 Term 1
Practice Problem Set 3
This is the suggested problems from the textbook for this week. You do not need
to hand in your work but you are encouraged to try all
MATH 2040 Linear Algebra II
2016-17 Term 1
Practice Problem Set 7
This is the suggested problems from the textbook for this week. You do not need
to hand in your work but you are encouraged to try all
MATH 2040 Linear Algebra II
2016-17 Term 1
Practice Problem Set 6
This is the suggested problems from the textbook for this week. You do not need
to hand in your work but you are encouraged to try all
MATH 2040 Linear Algebra II
2016-17 Term 1
Practice Problem Set 4
This is the suggested problems from the textbook for this week. You do not need
to hand in your work but you are encouraged to try all
MATH 2040 Linear Algebra II
2016-17 Term 1
Practice Problem Set 8
This is the suggested problems from the textbook for this week. You do not need
to hand in your work but you are encouraged to try all
MATH 2040 Linear Algebra II
2016-17 Term 1
Practice Problem Set 9
This is the suggested problems from the textbook for this week. You do not need
to hand in your work but you are encouraged to try all
MATH 2040 Linear Algebra II
2017-18 Term 1
Problem Set 2
due on September 29, 2017 at 5PM
Instructions: You are allowed to discuss with your classmates or seek help from the TAs
but you are required t
MATH 2040 Linear Algebra II
Lecture Notes by Martin Li
Lecture 9
Cayley-Hamilton theorem and existence of eigenvalues
1
In the last lecture, we have defined the important concept of invariant subspace
MATH 2040 Linear Algebra II
Supplementary Notes by Martin Li
Pre-requisite on complex numbers
1
For the purpose of algebra, the set of real numbers R is often not sufficient. For example, there
is no
MATH 2040 Linear Algebra II
2017-18 Term 1
Problem Set 5
due on November 10, 2017 at 5PM
Instructions: You are allowed to discuss with your classmates or seek help from the TAs
but you are required to
MATH 2040 Linear Algebra II
2017-18 Term 1
Problem Set 4
due on October 27, 2017 at 5PM
Instructions: You are allowed to discuss with your classmates or seek help from the TAs
but you are required to
MATH 2040 Linear Algebra II
Lecture Notes by Martin Li
Lecture 3
Span and linear independence
1
1
Linear combinations and span
Recall that a vector space consists of a set V together with two operatio
MATH 2040 Linear Algebra II
Lecture Notes by Martin Li
Lecture 14
Self adjoint and normal operators
1
In this section, we will study linear transformations between inner product spaces. Throughout thi
MATH 2040 Linear Algebra II
Lecture Notes by Martin Li
Lecture 5
Linear maps and their matrix representations
1
1
Linear maps and examples
Up to now we have defined vector spaces and some important co
MATH 2040 Linear Algebra II
2017-18 Term 1
Problem Set 6
due on November 29, 2017 at 5PM
Instructions: You are allowed to discuss with your classmates or seek help from the TAs
but you are required to
MATH 2040 Linear Algebra II
Lecture Notes by Martin Li
Lecture 8
Eigenvalues, eigenvectors and invariant subspaces
1
In previous lectures we have studied linear maps T : V W from a vector space V to a
MATH 2040 Linear Algebra II
Lecture Notes by Martin Li
Lecture 1
Vector spaces
1
1
Introduction
Linear algebra is the study of linear maps on finite dimensional vector spaces. This course serves as a
MATH 2040 Linear Algebra II
Lecture Notes by Martin Li
Lecture 13
Linear functionals
1
1
Linear functionals
Recall that we have studied in general linear transformations T : V W between two vector spa
MATH 2040 Linear Algebra II
2016-17 Term 1
Practice Problem Set 1
This is the suggested problems from the textbook for this week. You do not need
to hand in your work but you are encouraged to try all
Fundamental questions 1
4x2 + 3x3
1. Given a linear system x1
5x3
=2
= 0 , write down its coefficient matrix, solution
x5 = 0
vector, and augmented matrix.
2. Determine if the following matrices are
Fundamental questions 4
1. Show that B is the inverse of A, where
1 2
1 2
A=
,B =
1 1
1 1
2. If possible, find the inverse of the following matrices.
3 1
3 1
(a) A =
, (b) B =
2 2
6 2
3. Find the inve
Assignment 1
4x2 + 3x3
1. Given a linear system x1
5x3
=2
= 0 , write down its coefficient matrix, solution
x5 = 0
vector, and augmented matrix.
2. Determine if the following matrices are row equival
Assignment 2
1. Determine if the following matrices are reduced row-echelon form:
If yes, circle the leading 1s and indicate all the pivot columns; if no, explain the reason.
1 0 1 1 0 3
1 0 1 1
1 0 1
Assignment 4
1. Show that B is the inverse of A, where
1 2
1 2
A=
,B =
1 1
1 1
2. If possible, find the inverse of the following matrices.
3 1
3 1
(a) A =
, (b) B =
2 2
6 2
3. Find the inverse of the
2016 MAT2040
Classwork 3
Li Yiyao and Felix
1. Let A be a 3 5 matrix, B a 5 2 matrix, C a 3 4 matrix, D a 4 2 matrix, and E a
4 5 matrix. Determine which of the following matrix expressions exist and
Assignment 3
1. Use the symbol to form a chain on the following items:
(a) Matrices (b) Square matrices (c) Symmetric matrices (d) Diagonal matrices
2. Let A Mmn (R). Prove that
(i) A In = A
(ii)Im A
Assignment 2
1. Determine if the following matrices are reduced row-echelon form:
If yes, circle the leading 1s and indicate all the pivot columns; if no, explain the reason.
1 0 1 1 0 3
1 0 1 1
1 0 1
Assignment 6
1. Let V be a vector space. Show that the subspaces of V are vector spaces as well.
Solution. Let W be a subspace of V . Then
u + v W,
k u W,
u, v W
k R, u W
Also note that W is a sub
Assignment 3
1. Use the symbol to form a chain on the following items:
(a) Matrices (b) Square matrices (c) Symmetric matrices (d) Diagonal matrices
Solution.
Diagonal matrices Symmetric matrices Squa
Assignment 4
1. Show that B is the inverse of A, where
1 2
1 2
A=
,B =
1 1
1 1
Solution. Using the definition of an inverse matrix, you can show that B is the inverse
of A by showing that AB = I or I