Let us recall some remarkable examples of vector subspaces. The same observation applies to our examples of real vector spaces. Venkatesan guruswami the theory of errorcorrecting codes and more broadly, information theory, originated in claude shannons monumental worka mathematical theory of communication, published over 60 years ago in 1948. Our subject matter is intermediate calculus and linear algebra.
Here are some very well written notes on the gate mathematics topic linear algebra which were compiled by my friend suraj singh during his gate coaching at made easy institute in delhi. In short, these topics invite readers to see for themselves that linear algebra is a tool that a professional must have. Examples of vector spaces vector subspaces span, linear dependence, linear independence systems of linear equations bases overview of course this course is an introduction to linear algebra. In linear algebra we shall manipulate not just scalars, but also vectors, vector spaces, matrices, and linear transformations.
Taubes department of mathematics harvard university cambridge, ma 028 spring, 2010. Many applied problems from science, engineering and finance can be written in terms of linear algebra questions. Martin bright and daan krammer warwick, january 2011 contents 1 number systems and elds 3. These notes present a basic introduction to linear algebra with emphasis on few applications. It is used by the pure mathematician and by the mathematically trained scientists of all disciplines. In short, these topics invite readers to see for themselves that linear algebra. The lectures were distributed to the students before class, then posted on a. Or, so said alkhwarizmi, being the opening words of a 12th century latin translation of a work on arithmetic by alkhwarizmi ca. Lecture notes on linear algebra sissa people personal home. Call s linearly independent if it is not linearly dependent.
Given a basis e 1 e n for a vector space v, there is exactly one expression for. This book is directed more at the former audience than the latter, but it is hoped that the writing is su. The entry aij lies in the intersection of row i and. This set of notes is fairly self contained but there is enough algebra type problems arithmetic and occasionally solving equations that can show up that not having a good background in algebra.
Notes on linear algebra tuesday th december, 2016, 21. The set consisting of only the zero vector in a vector space v is a subspace of v, called the zero. So linear algebra studies these linear spaces and the maps between them that are compatible with the linear structure. I would like to thank him for numerous suggestions on the written notes. We begin by discussing the building blocks of linear algebra. Lecture notes on linear algebra department of mathematics.
These linear algebra lecture notes are designed to be presented as twenty ve, fty minute lectures suitable for sophomores likely to use the material for applications but still requiring a solid foundation in this fundamental branch. These notes of linear algebra course emphasize the mathematical rigour. From equation 2, we know that c is a 2 2 matrix with c 11 3 1 3 c 21 4 1 4 c 12 32 6 c. Gaussjordan elimination, matrix arithmetic, determinants, linear algebra, linear transformations, linear geometry, eigenvalues and eigenvectors. Systems of linear algebraic equations the main subject matter of this course are normally written as simple matrix equations of the form ax y. The basic operations that can be performed on matrices and that give rise to later uses and. Basic linear algebra in this chapter, we study the topics in linear algebra that will be needed in the rest of the book. F such that va i 1 a iv i empty sum is 0 write span fs. This module is a mixture of abstract theory, with rigorous proofs, and concrete calculations with matrices. Then we use our knowledge of matrices and vectors to develop a systematic procedure the gauss.
Preface these notes are accompanying a class on applied linear algebra math 4242 i am giving at the university of minneapolis in fall 2016 the website of the class is. These manipulations will include familiar operations such as addition. Ucb mathematics department of mathematics at university. The main idea of the course is to emphasize the concepts. This may sound somewhat abstract, and indeed, it is. I use other number systems, which i describe in the first section on rings. In this section well look at one of the fundamental algorithms of linear algebra called gaussjordan elimination. This is two equations and two variables, so as you know from high school algebra, you. Most students taking a course in linear algebra will have completed courses in di erential and integral calculus, and maybe also multivariate calculus, and will typically be secondyear students in university. It is customary to enclose the elements of a matrix in parentheses, brackets, or braces. We will use f to denote an arbitrary eld, usually r or c. Lecture notes for math 115a linear algebra fall of 2002 terence tao, ucla. We can use linear algebra to nd an exact formula for the nth term.
Intuitively, a vector space v over a eld f or an fvector space is a space with two operations. Lecture notes on linear algebra by david lerner department of mathematics university of kansas and the students of math 291 fall, 2007 these are notes of a course given in fall, 2007 to the honors section of our elementary linear algebra course. Because i wanted to make this a fairly complete set of notes for anyone wanting to learn linear algebra i have included some material that i do not usually have. Lecture 1 matrices, system of linear equations, elimination method. These gate maths notes are all hand written and will give you an overview of the syllabus as well as the key topics in linear algebra. The target reader is someone who is not necessarily a. I have kept the exposition lively and given an overall sense of breadth of application. These are the lecture notes for the course mth 415, applied linear algebra, a one semester class taught in 20092012. Most students taking a course in linear algebra will have completed courses in di erential and integral calculus, and maybe also multivariate calculus, and will. Since the course is an experimental one and the notes written. The equation for the eigenvalues for projections and re. Computational linear algebra okay, i hear what you are saying dude, enough with the theory talk, lets see some calculations. These linear algebra lecture notes are designed to be presented as twenty five.
We will see several such canonical form theorems in the notes. Geometrically, both of these equations describe lines in the xyplane and the existence of only one solution means that these two lines intersect in exactly one point. Linear algebra is one of the most applicable areas of mathematics. A matrix is a collection of numbers ordered by rows and columns. Here i denotes the identity matrix, the matrix having 1s on the main diagonal and. These linear algebra lecture notes are designed to be presented as twenty ve, fty minute lectures suitable for sophomores likely to use the material for applications but still requiring a solid foundation in this fundamental branch of mathematics. Warning for people trying to learn from these notes. Matrix algebra section 0 introduction to matrix algebra what you need to know already. Here are my online notes for my linear algebra course that i teach here at lamar. Brief introduction to logic and sets, brief introduction to proofs, basic linear. Create a new variable y by extracting a submatrix from x. You can still read a lot of the notes if you are just familiar with the real. These are the lecture notes for the course mth 415, applied linear algebra.
The rst portion of our workchapter 1 on in nite seriesmore properly belongs in the rst year, but is relegated to the second year by circumstance. Linear algebra as an introduction to abstract mathematics. Preface these notes started during the spring of 2002, when john majewicz and i each taught a section of linear algebra. These lecture notes correspond to the course linear algebra ii, as given at queen mary, university of london, in the. Chapter 1 introduces systems of linear equations, the gaussjordan method to. Linear algebra gabriel nagy mathematics department, michigan state university, east lansing, mi, 48824. Lecture notes for math 115a linear algebra terence tao. Introduction page 1 robertos notes on linear algebra chapter 4. Lecture 2 elementary matrices, invertible matrix, row reduction method. Joyos linear algebra notes joseph victor september 23, 20. Cramers rule consider a system of linear equations represented in matrix form as let a 1 be the matrix formed by replacing the 1st column of a with the vector, a 2 be the matrix formed by. Define a matrix x, then change x by performing the following. Martin bright and daan krammer warwick, january 2011 contents. A set with a linear structure in the sense of our discussion is called a linear space or vector space.
Linear algebra is the math of vectors and matrices. Call s linearly dependent if some v 2s depends on s nfvg, equivalently if there are distinct fv i g r 1. Linear algebra is strikingly similar to the algebra you learned in high school, except that in the place of ordinary single numbers, it deals with vectors. Chapter 1 introduction linear algebra is a branch of mathematics. Venkatesan guruswami the theory of errorcorrecting codes and more broadly, information theory, originated in claude shannons monumental worka mathematical theory. Linear algebra and markov chain methods are at the heart.
However, it is important to note that nearly all arguments in linear algebra use only the axioms for a eld and so are valid for any eld, which is why shall use a general eld kfor most of the course. Their comments and corrections have greatly improved the exposition. Print versions of the book either a physical copy or a pdf version have signi cant material available as supplements. These notes do assume that the reader has a good working knowledge of basic algebra. The power of linear algebra in practice stems from the fact that we can choose bases so as to simplify the form of the matrix representing the object in question. Notes on linear algebra queen marys school of mathematical. Geometrically, both of these equations describe lines in the xyplane and the existence of only one solution means that these two lines intersect in. In these notes we will explore the algebra and geometry of a certain family of very nice transformations. The next corollary summarises how transition matrices behave. Linear algebra ii lecture notes pdf 61p download book. Lecture notes for linear algebra pdf 268p these notes are intended for someone who has already grappled with the problem of constructing proofs. Many of the same algebraic operations youre used to performing on ordinary numbers a. Lecture notes math 43776308 advanced linear algebra i.
I have been tutoring math 51 for over three years and using linear algebra in advanced. The system of linear equations also known as a linear system. Multilinear algebra and tensor products lectures 914 14 chapter 2. This is also true of calculus, which is why these two fields are stressed in undergraduate mathematics education at ubc and other universities.
Made easy linear algebra download free notes, projects. Lecture notes on linear algebra pdf 220p download book. Multivariate statistics carey 82798 matrix algebra 1 introduction to matrix algebra definitions. Lecture notes on linear algebra david lerner department of mathematics university of kansas these are notes of a course given in fall, 2007 and 2008 to the honors sections of our elementary linear algebra course. This will also help us understand the adjective \ linear a bit better. Let nbe a positive integer and let r denote the set of real numbers, then rn is the set of all ntuples of real numbers.
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