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Complete Note for MATH0047 Advanced Linear Algebra
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MATH 0047Advanced Linear Algebra
Department of Mathematics
UCL
Lecture Notes
2021-2022
Isidoros Strouthos
November 21, 2021
, Abstract
In this course, we will aim to develop aspects of the theory of matrices and give an introduction to
the theory of vector spaces and the theory of linear maps. During the course, we are also due to see
examples of techniques which illustrate the use of the relevant mathematical objects.
Some of the topics we will cover in the first part of the course, apart from a revision of the basic
theory of matrices, will be row reduction and Gaussian elimination, elementary matrices, and matrix
determinants. The material will allow us to describe techniques used to find matrix inverses and
solve (systems of) linear equations.
The second part of the course will be an introduction to vector spaces and linear maps. It will include
the notions of inner products, subspaces, bases, orthogonality, norm, eigenvalues and eigenvectors,
results such as the Cauchy-Schwarz inequality, and techniques such as the Gram-Schmidt process.
,Contents
1 Matrices and linear equations 2
2 Determinants 55
3 Vector spaces 83
4 Linear maps 186
1
, Chapter 1
Matrices and linear equations
The mathematical objects we will be working with for most of this course are matrices. So, before
we go on to study some of the tools and techniques we will see in this course, let us review some
of the basic notions related to matrices, and introduce the subscript notation, which allows us to
prove, in a concise way, that matrices have various properties, by letting us “go down” to the level
of matrix entries, or numbers.
Definition 1.1. A matrix is a rectangular array of numbers.
In this course, our matrices will always contain complex numbers, so we may refer to them as
complex matrices, even though we shall often simply refer to them as matrices. So, unless otherwise
stated, below, a matrix is necessarily a complex matrix, i.e. it contains complex numbers as (array)
entries (though many of our examples will involve only real matrices, i.e. matrices with real numbers
as entries).
We will denote the set of complex numbers by C and the set of real numbers by R. So, x ∈ C means
that “x is a complex number”, while x ∈ R means that “x is a real number”.
If a matrix A has m rows and n columns, we will say that A has size m × n, or, equivalently, that A
is an m × n matrix
e.g.
A11 A12 ··· A1n
A21 A22 ··· A2n
A = .. is an m × n matrix
.. .. ..
. . . .
Am1 Am2 · · · Amn
2
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