Contents
Preface 4
Course summary 5
I Euclidean space 7
I.1 The vector space R2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
I.2 The vector spaces R3 and Rn . . . . . . . . . . . . . . . . . . . . . .
...
Contents
Preface 4
Course summary 5
I Euclidean space 7
I.1 The vector space R2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
I.2 The vector spaces R3 and Rn . . . . . . . . . . . . . . . . . . . . . . . . 10
I.3 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
II Systems of linear equations 19
II.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
II.2 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
II.3 Gaußian elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
II.4 Homogeneous and inhomogeneous SLEs . . . . . . . . . . . . . . . . . . 25
II.5 Inverting matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
III Vector spaces 27
III.1 Definition and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
III.2 Vector subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
III.3 Linear combination and span . . . . . . . . . . . . . . . . . . . . . . . . 30
III.4 Linear dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
III.5 Basis and dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
IV Inner product spaces 37
IV.1 Inner products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
IV.2 Orthogonality of vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
IV.3 Gram-Schmidt algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 41
V Linear transformations 43
V.1 Examples and general properties . . . . . . . . . . . . . . . . . . . . . . 44
V.2 Row and column spaces of matrices . . . . . . . . . . . . . . . . . . . . 45
V.3 Range and kernel, rank and nullity . . . . . . . . . . . . . . . . . . . . . 48
V.4 Orthogonal linear transformations . . . . . . . . . . . . . . . . . . . . . 51
VI Eigenvalues and eigenvectors 52
VI.1 Characteristic polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . 52
VI.2 Diagonalising matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
VI.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
VII Advanced topics 65
VII.1 Complex linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
VII.2 A little group theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
VII.3 Special relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Appendix 81
A Linear algebra with SAGE . . . . . . . . . . . . . . . . . . . . . . . . . 81
Index 84
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Preface
These are the lecture notes of the course Linear Algebra F18CF taught at Heriot-Watt
University to second year students. While the material covered in this course is itself
important for later courses, this course is mainly the first one to teach you how to prove
theorems. Linear Algebra is in fact the ideal course to learn this, as the proofs are rather
short, simple, and less technical than those in Analysis. Traditionally, switching from
algorithmic work to problem solving and proving theorems is very difficult for students,
and this course tries to ease the transition.
To understand theorems and proofs it is necessary to try to construct examples and/or
counterexamples to the statements. Playing and experimenting with definitions and theorems is one of the key activities for understanding mathematics. I therefore included an
appendix giving an introduction to doing Linear Algebra with the computer algebra programme SAGE. SAGE can be a valuable help in playing and experimenting, as it does most
of the tedious calculations for you.
Please note that these notes may still contain typos and other mistakes. If you should
find something that requires corrections (or if you have a good suggestion for improving
these notes), please send an email to
[email protected]. The lecture notes were created
relying on material from many different sources and are certainly not meant to be original.
Finally , I’d like to thank all students who spotted typos and took the time to let me
know, in particular Simone Rea.
Christian S¨amann
Remarks on Nomenclature and Notation
You should know what a Definition is. A Theorem is a mathematical (hopefully) profound and proved statement, and this is what we are most interested in. A Proposition
is a less profound statement, also coming with a proof. A Lemma is a proved auxiliary
statement1, usually used in the larger proof of a theorem. A Corollary is an interesting
consequence that follows almost immediately from a theorem.
A F marks additional reading material or questions to think about further. The end of
a proof is marked by . We use the convention that a := b and b =: a mean a is defined
as b. (In other sources, this is often written as a ≡ b.) Vectors are almost always labelled
by underlined characters x, v, u, ... While our vectors in Rn are always column vectors, we
also use row notation followed by a T for transpose to simplify typesetting. For example,
x = (1; 2; 3)T 2 R3. N∗, R∗ etc. denote the sets Nnf0g, Rnf0g etc. We label the set of
functions f : R ! R that are smooth, i.e. that can be differentiated infinitely many times,
by C1(R).
1Although there are many very deep statements that are usually called Lemma, see Wikipedia’s list of
Lemmata. Some people claim that a good lemma is worth a thousand theorems.
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Course summary
Outline
Euclidean Space. Vector space Rn, Matrices, Basic matrix operations, Determinants
Systems of Linear Equations. Gaußian elimination, Results on Homogeneous and
Inhomogeneous Linear Systems, Matrix Inversion
Vector Spaces. Definition and examples of Vector Spaces, Subspaces, Span, Linear
Independence, Bases and Dimension
Inner Product Spaces. Inner Products, Cauchy-Schwartz Inequality, Orthogonality,
Orthogonal Projection, Orthonormal Bases, Gram-Schmidt Process, Vector Products
Linear transformations. Row and Column Rank of a Matrix, Applications to Systems of Linear Equations, Range, Kernel, Rank and Nullity, Invertibility of Linear
Transformations, Linear Transformations and Matrices
Eigenvalues and Eigenvectors. Calculation of Eigenvalues and Eigenvectors, Symmetric Matrices, Diagonalisation of a Matrix, Cayley-Hamilton Theorem
Assessment and feedback
The final mark will be composed of 70% mark in final exam, 15% mark in midterm, 15%
marks in multiple-choice tests during the tutorials. You can use electronic calculators in
exams, however, the usual restrictions apply. The multiple-choice test will give you a good
feeling of how well you are progressing.
How to get the most out of this course
It is expected that you participate in both lectures and tutorials. For each hour spent
in a lecture, you should spend another hour at home to go through the material once
more. Do not expect to understand all the material immediately when it is presented in
class. Mathematics is difficult and requires considerable effort to learn. For this, it is also
necessary to try to solve the exercises on the tutorial sheets before coming to the tutorials.
The most effective and fun way to do this is to form small groups to work on the tutorials.
Make sure that you are able to explain the solutions to each other. Try to work continuously
throughout the term, this will make the preparation for the exam much easier for you. It
is very important that you keep a tidy set of lecture notes for yourself. Understanding the
material in class without lecture notes will be very difficult. As a backup, the lecture notes
required for the current tutorial will be made available on VISION when the tutorial is
handed out in class.
Recommended textbooks
This course is fairly self-contained and the material covered in the published lecture notes
is certainly sufficient to pass the exam. To deepen your knowledge in Linear Algebra, you
could use one of the following textbooks:
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Linear Algebra - Concepts and Methods by Martin Anthony and Michele Harvey.
Linear Algebra Done Right by Sheldon Axler, Springer.
Introduction to Linear Algebra by Gilbert Strang, Cambridge University Press.
Note that there is a wealth of lecture notes and other material freely available on the
internet. The book Linear Algebra in Schaum’s outline series is available on VISION.
By the end of the course, you should be able to...
B Use the Gaußian elimination procedure to determine whether a given system of simultaneous linear equations is consistent and if so, to find the general solution. Invert a
matrix by the Gaußian elimination method.
B Understand the concepts of vector space and subspace, and apply the subspace test to
determine whether a given subset of a vector space is a subspace.
B Understand the concepts of linear combination of vectors, linear (in)dependence, spanning set, and basis. Determine if a set of vectors is linearly independent and spans a
given vector space.
B Find a basis for a subspace, defined either as the span of a given set of vectors, or as
the solution space of a system of homogeneous equations.
B Understand the concept of inner product in general, and calculate the inner products
of two given vectors.
B Understand the concept of orthogonal projection and how to explicitly calculate the
projection of one vector onto another one. Use the Gram-Schmidt method to convert a
given basis for a vector space to an orthonormal basis. Understand (geometrically) the
concept of the vector product and calculate the vector product of two given vectors.
B Find the coordinates of a given vector in terms of a given basis - especially in the case
of an orthogonal or orthonormal basis.
B Calculate the rank of a given matrix and, from that, the dimension of the solution space
of the corresponding system of homogeneous linear equations. Calculate the determinant
of 2 × 2 and 3 × 3 matrices.
B Understand the concepts of linear transformation, range and kernel (nullspace). Understand the concept of invertibility of a linear transformation including injectivity and
surjectivity.
B Know and apply the Rank-Nullity Theorem.
B Compute the characteristic polynomial of a square matrix and (in simple cases) factorise
to find the eigenvalues.
B Determine whether a given square matrix is diagonalisable, and if so find a diagonalising
matrix.
B Apply the Cayley-Hamilton Theorem to compute powers of a given square matrix.
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