Math1048 linear algebra i university of southampton. Solving linear equations note 4 important suppose there is a nonzero vector x such that ax 0. Introduction to applied linear algebra stanford university. One novelty is that ive done as much linear algebra as possible over fields of nonzero characteristic. Students embarking on a linear algebra course should have a thorough knowledge of algebra, and familiarity with analytic geometry and trigonometry. My purpose with these notes is to introduce students to the concept of proof in a gentle manner. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes and rotations. The help that it gives to studentscomesfromtakingadevelopmentalapproachthisbookspresentation emphasizesmotivationandnaturalness,usingmanyexamples.
Here are my online notes for my linear algebra course that i teach here at lamar university. These notes do assume that the reader has a good working knowledge of basic algebra. A course for students with little or no background in linear algebra can focus on parts i and ii, and. Example let h 8 algebra to the case of nonlinear equations and maps. It also has some nice applications like linear algebra over nite elds and codes. We accept the currently acting syllabus as an outer constraint and borrow from the o. Kazdan topics 1 basics 2 linear equations 3 linear maps 4 rank one matrices 5 algebra of matrices 6 eigenvalues and eigenvectors 7 inner products and quadratic forms 8 norms and metrics 9 projections and re. I am assuming you met that material in a multivariable calculus course. Notes on linear algebra tuesday th december, 2016, 21. And, of course, many topics in linear algebra are discussed with varying degrees of.
We begin our study of linear algebra with an introduction and a motivational example. It is intended for a student who, while not yet very familiar with abstract reasoning, is willing to study more rigorous mathematics than what is presented in a \cookbook style calculus type course. A set with a linear structure in the sense of our discussion is called a linear space or vector space. Department of pure mathematics and mathematical statistics. Many elementary proofs are omitted, but the experienced reader will have no di.
Recompute each of these three costs and notice that the computations are linear in character. Mostly there are two themes in this collection of short handouts. Jiwen he, university of houston math 2331, linear algebra 7 14. Exercises and problems in linear algebra portland state university. Here, for example is an article describing how to do linear regression. Matrix algebra of some sample statistics variance of a linear combination variancecovariance matrix of several linear combinations covariance matrix of two sets of linear combinations matrix algebra of sample statistics james h.
Linear algebra vectors matrices points as vectors geometric transformations in 2d homogeneous coordinates cse 167, winter 2018 2. Contribute to zlotusnoteslinearalgebra development by creating an account on github. The reason for this is that linear structures abound in mathematics, and so linear algebra has applications everywhere see below. Recipe flour cups sugar tablespoons pancakes 2 1 wa. In particular, those by gilbert strang are particularly easy to read and understand. This is the point of view of this book, more than a presentation of linear algebra for its own sake. Jones begin the abstract here, typewritten and doublespaced. Department of pure mathematics and mathematical statistics, university of cambridge. As examples, the two matrices below do not have inverses a 1. Linear algebra is well suited for this, as it has a number of beautiful but elementary and easy to prove theorems. Linear algebra math 3333 04 spring 2011 final exam. Linear maps on vector spaces are the basis for a large area of mathematics, in particular linear equations and linear differential equations, which form the basic language of the physical sciences.
This subspace came fourth, and some linear algebra books omit itbut that misses the beauty of the whole subject. 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. This may sound somewhat abstract, and indeed, it is. It can be shown that the range of l is the entire space cr while the kernel of l is spanned by the functions sinx and cosx.
The material is developed completely from scratch, but at a faster pace than a beginning linear algebra course. Linear algebra provides a way of compactly representing and operating on sets of linear equations. Designed for a onesemester linear algebra course at the undergraduate level, the book gives instructors the option of tailoring the course. Several of the numerical examples in this section are adapted from strangs linear algebra and its applications, second edition academic press, 1980. Linear algebra is central to almost all areas of mathematics. Linear algebra a pure mathematical approach, is a nice companion to these notes. Linear algebra example problems vector space basis. Another great example of a realworld problem where linear algebra proves to be very effective is the problem of data compression, that is. We shall develop the material of linear algebra and use it as setting for the relevant material of intermediate calculus. Lets start with a simple example that shows what linear algebra is about. The lecture notes correspond to the course linear algebra and di. 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. 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. It is assumed that the reader has already studied matrix algebra or linear algebra, however, these notes are completely selfcontained.
Thus, under the second sense described above, a spanning set is minimal if and only if it contains no vectors that are linear combinations of the others in that set. For each question, to receive full credit you must show all work. The paper is mostly focused on resultants of nonlinear maps. Applications version 1 by howard anton and chris rorres and linear algebra and its applications 10 by gilbert strang are loaded with applications. In rn the row space and nullspace have dimensions r.
A matrix in which each entry is zero is called a zeromatrix, denoted by 0. The great majority of the results in beginning linear and multilinear are straightforward and can be veri ed by the thoughtful student. Much of it is simple vector and matrix algebra that you can learn from the summary itself, particularly if you devise and work through enough two and threedimensional examples as. Linear algebra class notes fraleigh and beauregard copies of the classnotes are on the internet in pdf format as given below. Dont be impatient and dont just try to get it over with. Linear algebra is the study of linear transformations and their algebraic properties. It is assumed that the reader is already familiar with this subject, if not with the exact terminology and notation used here. Systems of linear equations are common in science and mathematics. The lemma says that if we have a spanning set then we can remove a to get a new set with the same span if and only if is a linear combination of vectors from. So linear algebra studies these linear spaces and the maps between them that are compatible with the linear structure.
This set of notes is an activityoriented introduction to the study of linear and multilinear algebra. Linear algebra with applications open edition be a champion of open educational resources. Linear algebra notes fraleigh and beauregard webpage. Dorier, robert, robinet and rogalski in their paper 5 give various examples of a group of their students work as testimony to the necessity and the importance of set theory knowledge in responding accurately to linear algebra questions. Linear algebra is one of the most applicable areas of mathematics. For sections 1 and 2, no explanations are necessary. This is two equations and two variables, so as you know from high school algebra, you. For example, consider the following system of equations.
Linear algebra linear transformations transformation linear. First steps are described in direction of mandelbrotset theory, which is direct extension of the eigenvalue problem from linear algebra. Examples in two and three dimensions can be used to provide valuable insight into important concepts of this course. In the first, we construct a matrix and perform row operations to show that we obtain a pivot in each column. Consider the following recipes for pancakes and wa. A geometry toolbox, third edition teaches undergraduatelevel linear algebra in a comprehensive, geometric, and algorithmic way. And for those more interested in applications both elementary linear algebra. I know this material like the back of my hand, and ive worked with such a huge number of diverse students that i feel like know where students get confused, and how students succeed. First, that rather than think of an m by n matrix as a doublyindexed array, it is often more enlightening to think of it as a ntuple of columns which are, of course, mvectors or an mtuple of rows. Give examples of 2 2 matrices that commute and ones that dont. Look for a linear algebra quick reference sheet at the sage website.
Despite the fact that these are my class notes they should be accessible to anyone wanting to learn linear algebra or needing a refresher. Two different methods are used to check for linear independence of the vectors. I have been tutoring math 51 for over three years and using linear algebra in advanced mathematics and computer science course work and as an software engineer. This example is a hint at a much bigger idea central to the text. For example, z and r are infinite sets of different cardinalities while z and q are infinite sets of the same cardinality. The use of linear algebra in modeling the probabilities of predicted future occurrences singular value decomposition svd and similar methods can be used to factor matrices into subspaces which describe their behavior. Three people denoted by p1, p2, p3 intend to buy some rolls, buns, cakes and bread. Our subject matter is intermediate calculus and linear algebra. Contribute suggestions for improvements,new content, or errata. Heres just a fraction of what you can do with linear algebra.
Through many examples and realworld applications, practical linear algebra. It pays to spend some time just making sure that you have the main concepts clear in your head. This module restricts itself to the vector space rn to build an intuitive understanding of the concepts of linear algebra and tools for calculations. However, much of the power of linear algebra will be the ability to work with flat or straight objects in higher dimensions, without concerning ourselves with visualizing the situation. Several simple realworld applications of linear algebra tools. Fundamentals of linear algebra and optimization penn engineering. Matrices with examples and questions with solutions. Manin linear algebra and geometry in the series of algebra logic and applications, gordonbreach 1989. Linear algebra is essential in analysis, applied math, and even in theoretical mathematics. 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. Download pdf a first course in linear algebra university of.
Linear geometry also involves higher dimensional analogues of lines, for examples, lines and planes in 3space, or r3 as we will denote it. It is used by the pure mathematician and by the mathematically trained scientists of all disciplines. A thesis abstract should consist of 350 words or less including the heading. In practice, i confine myself to examples over the integers mod 2, 3, and 5, but i think this is enough to get the point across. The examples, exercises, and proofs files are pdf files prepared in beamer and they contain worked examples and exercises and a few proofs which are not in the regular classnotes. The archetypes are available in a pdf form which could be used as a workbook. Dual spaces friday 3 november 2005 lectures for part a of oxford fhs in mathematics and joint schools linear functionals and the dual space dual bases annihilators an example the second dual. Preface here are my online notes for my linear algebra course that i teach here at lamar university. This work is licensed under a creative commons attributionnoncommercialsharealike 3. These notes are for a graduate course in linear algebra. Examples of scalars are temperature, distance, speed, or mass all quantities that have a magnitude but no direction, other than. First steps are described in direction of mandelbrotset theory, which is direct extension of the eigenvalue problem from linear algebra, and is related by. When a linear transformation is described in term of a matrix it is easy to determine if the linear transformation is onetoone or not by checking the linear dependence of. Access study documents, get answers to your study questions, and connect with real tutors for mat 1503.
Mathematics department home page infoeagle home page boston college home page. My main goal in writing these notes was to give to the student a concise overview of the main concepts,ideas and results that usually are covered in the rst course on. Steiger department of psychology and human development vanderbilt university p3, 2010. Indeed, that is the main point of these notes to convince the beginner that the subject is accessible. Linear algebradefinition and examples of linear independence. If it is not linear, give an example that shows why not. Linear algebra uses vectors to study systems of linear equations. Linear spaces this chapter is a brief survey of basic linear algebra. A vector is a quantity with a magnitude and a direction. This powerful science is based on the notions of discriminant hyperdeterminant and resultant, which today can be e. Jiwen he, university of houston math 2331, linear algebra 16 6. The following resources are not endorsed by the university of cambridge. This is why there are numerous applications, some fairly.
So we have a vector space of functions, and the methods of linear algebra are used to analyze the experiment. David cherney, tom denton, rohit thomas and andrew waldron. Try the following on your own, then use the book and notes where you need help. A course for students with little or no background in linear algebra can focus on parts i and ii, and cover just a few of the more advanced applications in part iii. Example 1 the 2 by 2 matrix a 1 2 1 2 is not invertible. These notes of linear algebra course emphasize the mathematical rigour over the applications, contrary to many books on linear algebra for engineers. It supposed to be a rst linear algebra course for mathematically advanced students. R n by giving specific numeric examples for which the vectors sum to a vector in. Linear algebra math 3333 04 spring 2011 final exam practice problem solutions instructions. C10 robert beezer in example tmp the rst table lists the cost per kilogram to manufacture each of the three varieties of trail mix bulk, standard, fancy.
There are several things about these linear algebra notes that are a little unusual. Eigenvalues and eigenvectors questions with solutions. The main lesson of an introductory linear algebra course is this. This is why there are numerous applications, some fairly unusual. Vectors represent magnitude and direction in multiple dimensions examples translation of a point. Despite the fact that these are my class notes, they should be accessible to anyone wanting to learn linear algebra or needing a refresher. We do not assume that calculus is a prerequisite for this course, but we do include examples and exercises requiring calculus in the text. No calculators, books or notes of any form are allowed. You may refer to theorems in the book or from class unless the question speci.