FACULTY OF SCIENCE
SCHOOL OF MATHEMATICS AND
STATISTICS
MATH2601
HIGHER LINEAR ALGEBRA
Session 2, 2007
2
MATH2601 Course Outline
Information about the course
Course Authority and lecturer: Dr Astrid an Huef, Room RC-6108, phone
9385 7088, email astrid@uns
4. The Determinant
This section explains one rigorous way of defining the determinant of a square matrix.
The idea is to get a feel for this definition and how it leads to fairly straightforward
proofs of the most important properties of the determinant.
2. The structure of vector spaces and linear transformations of
vector spaces
Let V be a vector space over a field F.
Definition
2.1. A linear combination of vectors v 1 , v 2 , . . . , v n V is a vector of the
Pn
form i=1 ci v i where the ci are scalars
School of Mathematics and Statistics
The University of New South Wales
MATH2601
Higher Linear Algebra
Semester 1, 2015
Short writing assignment Sample solutions and comments
Question (a):
Let T : C3 C3 be the map which performs multiplication by the matri
3. Inner products, and linear transformations on inner product spaces
and their adjoints
Let V be a vector space over F = R or C. An inner product on V is a function
h | i : (v, w) 7 hv | wi of V V into F such that
(IP1)
(IP2)
(IP3)
(IP4)
hv | vi is a non
7. The exponential of a matrix
Consider a system of ordinary differential equations
x01 (t) = a11 x1 (t) + a12 x2 (t) + b1 (t)
x02 (t) = a21 x1 (t) + a22 x2 (t) + b2 (t)
where aij R and bi : R R are given and
this system in vector form x0 = Ax + b where
x
5. Diagonalisation
Given a linear transformation T : V V of a finite-dimensional vector space V ,
when is there a basis B = cfw_v 1 , . . . , v n for V such that the matrix AT = (aij ) of T
with respect to B is diagonal? The key observation is that AT is
School of Mathematics and Statistics
The University of New South Wales
MATH2601
Higher Linear Algebra
Semester 1, 2015
Short writing assignment (worth 5%)
This assignment is due at the end of the lecture on Tuesday in Week 9. Late assignments
will not be
6. The Jordan Canonical Form
A matrix A Mn (C) of the form
1
1
.
.
.
. 1
is called a Jordan block . We say A is in Jordan form if A is a direct sum of Jordan
blocks. In this chapter we will show that every matrix A Mn (C) is similar to one
in Jordan for
Permutations
Aim lecture: Intro algebra of permutations as a tool to study determinants.
Defn
Let Jn = cfw_1, . . . , n.
Let Fn the set of all functions of form f : Jn Jn .
We denote f Fn by [f (1) f (2) . . . f (n)].
A permutation is some Fn which is inv
Lecture 0: Boring admin stuff
Daniel Chan
UNSW
Semester 2 2013
Daniel Chan (UNSW)
Lecture 0: Boring admin stuff
Semester 2 2013
1/7
MATH2601
Higher linear algebra
Credit: 6 Units of Credit (6UOC).
Prerequisites: MATH1231 or MATH1241 or MATH1251 each with
Composite of linear maps
Aim lecture: Linear maps can be combined via composition. We introduce this
concept here.
Prop
Let S, T : V V 0 , S 0 , T 0 : V 0 V 00 be F-lin maps.
1
2
3
The composite fn T 0 T : V V 00 defined by (T 0 T )v = T 0 (T v) is
F-line
Dual vector space
Aim lecture: We generalise the notion of transposes of matrices to arbitrary
linear maps by introducing dual vector spaces.
In most of this lecture, we allow F to be a general field.
Defn
Let V = F-space. The dual of V is the F-space V =
Old & new co-ordinates
Aim lecture: A wise choice of co-ordinates can make life much easier. We give
some examples showing how to make a linear change of co-ords to facilitate
calculations.
Suppose we have old co-ords (x1 , . . . , xn ) Fn & some new co-o
List of properties
Here are some additional properties of the determinant function.
Prop
Throughout let A, B Mnn .
1
If A = (aij ) is upper triangular then det(A) = a11 a22 . . . ann .
2
If a row or column of A is 0, so is det(A).
3
det(AB) = det(A) det(B
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 enough, an F-space is n dimensional if it is isomorphic
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 with identity e & g G . For n Z+ we define
g n = g g . .
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 we have a) T (v + v0 ) = T v + T v0 & b)
T (v) = T v i
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)
Lecture 5: The determinant
Semester 2 2012
1 / 10
Even & o
MATH2601: Higher Linear Algebra. Semester 1, 2015
Skeleton notes by Astrid an Huef, edited by Catherine Greenhill
These skeleton notes will bear some resemblance to what happens in class, but
are in no way a replacement for attending class. Please let the