# MA211: Linear Algebra

## Course Syllabus for "MA211: Linear Algebra"

This course is an introduction to linear algebra.  It has been argued that linear algebra constitutes half of all mathematics.  Whether or not everyone would agree with that, it is certainly true that practically every modern technology relies on linear algebra to simplify the computations required for Internet searches, 3-D animation, coordination of safety systems, financial trading, air traffic control, and everything in between. Linear algebra can be viewed either as the study of linear equations or as the study of vectors.  It is tied to analytic geometry; practically speaking, this means that almost every fact you will learn in this course has a picture associated with it.  Learning to connect the facts with their geometric interpretation will be very useful for you. The book which is used in the course focuses both on the theoretical aspects as well as the applied aspects of linear algebra.  As a result, you will be able to learn the geometric interpretations of many of the algebraic concepts in this subject.  Additionally, you will learn some standard techniques in numerical linear algebra, which allow you to deal with matrices that might show up in applications.  Toward the end, the more abstract notions of vector spaces and linear transformations on vector spaces will be introduced. In college algebra, one becomes familiar with the equation of a line in two-dimensional space: y = mx+b.  Lines can be generalized to planes and “hyperplanes” in many-dimensional space; these objects are all described by linear relations.  Linear transformations are ways of rotating, dilating, or otherwise modifying the underlying space so that these linear objects are not deformed.  Linear algebra, then, is the theory and practice of analyzing linear relations and their behavior under linear transformations.  According to the second interpretation listed above, linear algebra focuses on vectors, which are mathematical objects in many-dimensional space characterized by magnitude and direction.  You can also think of them as a string of coordinates.  Each string may represent the state of all the stocks traded in the DOW, the position of a satellite, or some other piece of data with multiple components.  Linear transformations change the magnitude and direction of vectors—they transform the coordinates without changing their fundamental relationships with one another.  Linear transformations are often written in a compact and easily-readable way by using matrices. Linear algebra may at first seem dry and difficult to visualize, but it is one of the most useful subjects you can learn if you wish to become a business-person, a physicist, a computer-programmer, an engineer, or a mathematician. Remember, the prerequisite of this course is one variable calculus course and a reasonable background in college algebra.

### Learning Outcomes

Upon successful completion of this course, you will be able to:

• define and identify linear equations;
• write a system of equations in matrix-vector form;
• explain the geometric interpretation of a system of linear equations;
• define and distinguish between singular and nonsingular matrices, and calculate a matrix inverse;
• relate invertibility of matrices to solvability of linear systems;
• define and characterize Euclidean space;
• define and compute dot and cross-products;
• define and identify vector spaces and subspaces;
• define spanning set and determine the span of a set of vectors;
• define and verify linear independence;
• define basis and dimension;
• show that a set of vectors is a basis;
• define and compute column space, row space, nullspace, and rank;
• define and identify isomorphisms and homomorphisms;
• use row and column space to solve linear systems;
• state the rank-nullity theorem;
• define inner product, inner product space, and orthogonality;
• interpret inner products geometrically;
• state the properties of determinants;
• compute the determinant using cofactor expansions, row reduction, and Cramer’s Rule;
• define and compute the characteristic polynomial of a matrix;
• define and compute eigenvalues and eigenvectors;
• explain the geometric significance of eigenvalues and eigenvectors;
• define similarity and diagonalizability;
• identify similar matrices;
• identify some necessary conditions for diagonalizability;
• find LU factorization of a matrix and solve systems of equations using this factorization;
• construct a simplex tableau;
• find a PLU factorization of a matrix and solve systems of equations using PLU factorization;
• find the QR factorization of a matrix;
• use the Gram Schmidt process to find an orthonormal basis;
• use iterative methods to approximate eigenvalues;
• define and provide examples of inner product spaces;
• state and prove the Cauchy Schwarz inequality;
• apply the Gram Schmidt process to inner product spaces;
• find matrix representations for a given linear transformation;
• find the range and kernel of a transformation;
• use linear transformations to prove that vector spaces are isomorphic; and
• solve least squares problems.

### Course Requirements

In order to take this course you must:

√    Have continuous broadband Internet access.

√    Have the ability/permission to install plug-ins or software (e.g., Adobe Reader or Flash).

√    Have the ability to download and save files and documents to a computer.

√    Have the ability to open Microsoft files and documents (.doc, .ppt, .xls, etc.).

√    Be competent in the English language.

√    Have read the Saylor Student Handbook.

### Course Information

Welcome to MA211.  Below, please find general information on the course and its requirements.

Primary Resources: This course is composed of a range of different free, online materials. However, the course makes primary use of the following materials:

Requirements for Completion: In order to complete this course, you will need to work through each unit and all of its assigned materials.  You will also need to complete assignments for Unit 1 through Unit 6 as well as the final exam.

Note that you will only receive an official grade on the Final Exam.  However, in order to adequately prepare for this exam, you will need to work through the assignments and all the reading material in the course.

In order to “pass” this course, you will need to earn a 70% or higher on the Final Exam.  Your score on the exam will be tabulated as soon as you complete it.  If you do not pass the exam, you may take it again.

Time Commitment: This course should take you a total of 135.75 hours to complete.  Each unit includes a “time advisory” that lists the amount of time you are expected to spend on each subunit.  These should help you plan your time accordingly.  It may be useful to take a look at these time advisories and to determine how much time you have over the next few weeks to complete each unit, and then to set goals for yourself.  For example, Unit 1 should take you 25.5 hours.  Perhaps you can sit down with your calendar and decide to complete all of Unit 0 (a total of 1.25 hours) and half of subunit 1.1 (a total of 3 hours) on Monday night; the rest of subunit 1.1 (a total of 3 hours) on Tuesday; half of subunit 1.2 (a total of 3 hours) on Wednesday; etc.

Tips/Suggestions: It will likely be helpful to have a calculator on hand for this course.

Make sure you have a solid understanding of the pre-requisite topics outlined in Unit 0 before moving on to other units in the course.  As you read, take careful notes on a separate sheet of paper.  Mark down any important equations, formulas, and definitions that stand out to you.  It will be useful to use this “cheat sheet” as a review prior to completing the Final Exam.

### Preliminary Information

• Open Textbook Challenge Winner: Elementary Linear Algebra

*Elementary Linear Algebra was written and submitted to the Open Textbook Challenge by Dr. Kenneth Kuttler of Brigham Young University.  Dr. Kuttler wrote this textbook for use by his students at BYU.  According to the introduction of Elementary Linear Algebra, “this is intended to be a first course in linear algebra for students who are sophomores or juniors who have had a course in one variable calculus and a reasonable background in college algebra.”  A solutions manual for the textbook is included.*

• Some Prerequisite Topics

To learn linear algebra, you must have knowledge of some topics from elementary algebra.  In this unit, you will review these topics.  Specifically, the unit begins with a brief review of set notation and then moves to a review of complex numbers.  This is especially important in linear algebra, since polynomial equations with real coefficients often have complex numbers as roots.  The quadratic formula, which is the next topic you will review, is a good example of this phenomenon.

This unit should take you approximately 1.25 hours to complete.

***☐    ***Subunit 0.1: 0.5 hour

☐    Subunit 0.2: 0.5 hour

☐    Subunit 0.3: 0.25 hour

• Sets and Functions

• Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 1: Some Prerequisite Topics:” “Section 1.1: Sets and Set Notation,” “Section 1.2: Functions,” and “Section 1.3: Graphs of Functions” The Saylor Foundation does not yet have materials for this portion of the course. If you are interested in contributing your content to fill this gap or aware of a resource that could be used here, please submit it here.

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• Complex Numbers

• Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 1: Some Prerequisite Topics:” “Section 1.4: The Complex Numbers,” “Section 1.5: Polar Form of Complex Numbers,” and “Section 1.6: Roots of Complex Numbers” The Saylor Foundation does not yet have materials for this portion of the course. If you are interested in contributing your content to fill this gap or aware of a resource that could be used here, please submit it here.

Submit Materials