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Linear Algebra
Purpose of Course showclose
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.
Course Information showclose
Primary Resources: This course is composed of a range of different free, online materials. However, the course makes primary use of the following materials:
- Kenneth Kuttler’s Elementary Linear Algebra
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.
Learning Outcomes showclose
- 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 showclose
√ Have access to a computer.
√ 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.
Preliminary Information
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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.
Elementary Linear Algebra (PDF)
Elementary Linear Algebra Solutions Manual (PDF)
Elementary Linear Algebra (iTunes) -
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.
Time Advisory show close
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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”
Link: 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” (PDF)
Instructions: Please click on the link above, and read the indicated sections on pages 11–15. Section 1.1 briefly discusses set notation; Sections 1.2 and 1.3 provide a brief review of functions. This reading should take approximately 30 minutes to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.See a broken link? Please let us know!
- 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”
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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”
Link: 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” (PDF)
Instructions: Please click on the link above, and read the indicated sections on pages 15-20. These sections will provide a review of complex numbers.
This reading should take approximately 30 minutes to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.See a broken link? Please let us know!
- 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”
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The Quadratic Formula
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 1: Some Prerequisite Topics:” “Section 1.7: The Quadratic Formula”
Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 1: Some Prerequisite Topics:” “Section 1.7: The Quadratic Formula” (PDF)
Instructions: Please click on the link above, and read Section 1.7 on pages 20 and 21. This section will provide a review of the quadratic formula.
This reading should take approximately 15 minutes to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.See a broken link? Please let us know!
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 1: Some Prerequisite Topics:” “Section 1.7: The Quadratic Formula”
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Unit 1: Vector Products, Systems of Equations, and Matrices
This unit begins with a review of vectors. You will learn the geometric meaning of vectors, which is especially significant in R2 and R3. Next, you will learn the geometric meaning of vector addition and scalar multiplication. Finally, you will apply study vectors in the context of physics to model force and other physical vectors like velocity.
Time Advisory show close
In the next chapter, you will begin learning about vector products. There are two ways of multiplying vectors, both of which are of great importance in applications. The first type of a product is called the dot product, also called the scalar product or the inner product. You will then study the geometric significance of the dot product and applications of dot product by studying the concepts of work and projections. Next, you will begin the study of cross products. The cross product is the other way of multiplying vectors, and it is different from the dot product in fundamental ways. You will learn both the geometric meaning of the cross product and the description in terms of coordinates. Both descriptions of the cross product are important; the geometric description is necessary to understand the applications to physics and geometry while the coordinate description is necessary to actually compute the cross product. You will then learn techniques, which will allow you to discover vector identities and simplify expressions involving cross and dot products in three dimensions.
Next, you will begin exploring systems of linear equations. The basic idea is to study situations where there are several different variables that are related in multiple ways. These linear equations could describe budget constraints in a business, physical constraints in an engineering problem, or any number of other situations. The key is that these constraints can be described by linear equations. The geometric interpretation of these constraints is that each equation describes a line or plane where potential solutions to the problem must lie. The task then is to figure out what combination of variable values solves all of the different linear equations at the same time. Geometrically, this is where all of the lines or planes intersect. Just as is the case with the problems that the equations may be modeling, the system of equations will sometimes have no solution, will sometimes have a single solution, and will sometimes have an infinite number of solutions. Finally, you will learn about matrices andhow to write a system of linear equations as a matrix equation. While this may have at first appeared to be merely a way of putting your coefficients in a table, matrices in fact have many interesting (but not immediately obvious!) properties.
Learning Outcomes show close
- 1.1 Fn and Vector Products
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1.1.1 Vectors and their Geometric Meaning
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 2: Fn:” “Section 2.1: Algebra in Fn,” “Section 2.2: Geometric Meaning of Vectors,” and “Section 2.3: Geometric Meaning of Vector Addition”
Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 2: Fn:” “Section 2.1: Algebra in Fn.” “Section 2.2: Geometric Meaning of Vectors,” and “Section 2.3: Geometric Meaning of Vector Addition” (PDF)
Instructions: Please click on the link above, and read the indicated sections on pages 23–27. Section 2.1 introduces you to algebraic operations done with elements of Fn. In Section 2.2, you will explore the geometric meaning of vectors, and in Section 2.3, you will study the geometric interpretation of vector addition. These readings should take you approximately 1 hour to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.See a broken link? Please let us know!
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 2: Fn:” “Section 2.1: Algebra in Fn,” “Section 2.2: Geometric Meaning of Vectors,” and “Section 2.3: Geometric Meaning of Vector Addition”
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1.1.2 Length of a Vector and Scalar Multiplication
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 2: Fn:” “Section 2.4: Distance between Points in Length of a Vector” and “Section 2.5: Geometric Meaning of Scalar Multiplication”
Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 2: Fn.” “Section 2.4: Distance between Points In Rn Length of a Vector” and “Section 2.5: Geometric Meaning of Scalar Multiplication” (PDF)
Instructions: Please click on the link above, and read the indicated sections on pages 27–31. In Section 2.4, you will study how distance is defined between two points in Rn. In Section 2.5, you will explore the geometric meaning of scalar multiplication. These readings should take you approximately 1 hour to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.See a broken link? Please let us know!
- Assessment: Professor Kenneth Kuttler’s Elementary Linear Algebra: "Chapter 2:" “Section 2.6: Exercises”
Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 2: Fn.” “Section 2.6: Exercises” (PDF)
Instructions: Please click on the link above to open the PDF. Scroll down to page 31 to Section 2.6, and complete problems 1, 2, 4, and 5. Next, click on “Solutions” (PDF) and check your answers on pages 5–6. This assessment should take you approximately 1 hour to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.See a broken link? Please let us know!
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 2: Fn:” “Section 2.4: Distance between Points in Length of a Vector” and “Section 2.5: Geometric Meaning of Scalar Multiplication”
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1.1.3 Vectors and Physics
- Reading: Professor Kenneth Kuttler's Elementary Linear Algebra: "Chapter 2: Fn." "Section 2.7: Vectors and Physics"Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 2:” “Section 2.7: Vectors and Physics” (PDF)
Instructions: Please click on the link above, and read Section 2.7 on pages 32–36. In this section, you will learn about the concept of force. This reading should take you approximately 1 hour to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.See a broken link? Please let us know!
- Assessment: Professor Kenneth Kuttler's Elementary Linear Algebra: "Chapter 2: Fn." "Section 2.8: Exercises"
Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 2: Fn." “Section 2.8: Exercises”(PDF)
Instructions: Please click on the link above to open the PDF. Scroll down to page 36, and complete problems 2, 3, 7, 9, 10, and 11. Next, click on “Solutions” (PDF) and check your answers on pages 6–8. This assessment should take you approximately 2 hours to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.See a broken link? Please let us know!
- Reading: Professor Kenneth Kuttler's Elementary Linear Algebra: "Chapter 2: Fn." "Section 2.7: Vectors and Physics"
- 1.2 Vector Products
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1.2.1 The Dot Product
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 3: Vector Products:” “Section 3.1: The Dot Product” and “Section 3.2: The Geometric Significance of the Dot Product”
Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 3: Vector Products:” “Section 3.1: The Dot Product” and “Section 3.2: The Geometric Significance of the Dot Product” (PDF)
Instructions: Please click on the link above, and read the indicated sections on pages 39–47. Section 3.1 will provide the definition and properties of the dot product. Section 3.2 will discuss the geometric meaning of the dot product and then apply the ideas to the concepts of work and projection. These readings should take you approximately 1 hour and 30 minutes to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.See a broken link? Please let us know!
- Assessment: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 3: Vector Products:” “Section 3.3: Exercises”
Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 3: Vector Products:” “Section 3.3: Exercises” (PDF)
Instructions: Please click on the link above to open the PDF. Scroll down to page 47, and work on problems 1, 2, 4, 6, 10, 14, 15, 17, 20, and 21. Next, click on “Solutions” (PDF) and check your answers on pages 8–10. This assessment should take you approximately 2 hours and 30 minutes to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.See a broken link? Please let us know!
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 3: Vector Products:” “Section 3.1: The Dot Product” and “Section 3.2: The Geometric Significance of the Dot Product”
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1.2.2 The Cross Product
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 3: Vector Products:” “Section 3.4: The Cross Product”
Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 3: Vector Product:” “Section 3.4: The Cross Product” (PDF)
Instructions: Please click on the link above, and read Section 3.4 on pages 48–54. Section 3.4 will provide the definition, properties, and the geometric meaning of the cross product. This reading should take you approximately 1 hour to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.See a broken link? Please let us know!
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 3: Vector Products:” “Section 3.4: The Cross Product”
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1.2.3 The Vector Identity Machine
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 3: Vector Products:” “Section 3.5: The Vector Identity Machine”
Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 3: Vector Products:” “Section 3.5: The Vector Identity Machine” (PDF)
Instructions: Please click on the link above, and read Section 3.5 on pages 54–56. Section 3.5 will introduce a technique that will allow you to discover vector identities and simplify expressions involving cross and dot products in three dimensions. This reading should take you approximately 30 minutes to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.See a broken link? Please let us know!
- Assessment: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 3: Vector Products:” “Section 3.6: Exercises”
Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 3: Vector Products:” “Section 3.6: Exercises” (PDF)
Instructions: Please click on the link above to open the PDF. Scroll down to page 56, and complete problems 1, 4, 6, 7, 8, 9, 10, 13, 16, 18, and 20. Next, click on “Solutions” (PDF) and check your answers on pages 11–14. This assessment should take you approximately 3 hours and 30 minutes to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.See a broken link? Please let us know!
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 3: Vector Products:” “Section 3.5: The Vector Identity Machine”
- 1.3 Systems of Equations
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1.3.1 Systems of Equations, Geometry
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 4: Systems of Equations:” “Section 4.1: Systems of Equations, Geometry”
Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 4: Systems of Equations:” “Section 4.1: Systems of Equations, Geometry” (PDF)
Instructions: Please click on the link above, and read Section 4.1 on pages 59–61. Section 4.1 will explore how to find solution(s) for a system of linear equations by graphing. This reading should take you approximately 30 minutes to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.See a broken link? Please let us know!
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 4: Systems of Equations:” “Section 4.1: Systems of Equations, Geometry”
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1.3.2 Systems of Equations, Algebraic Procedures
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 4: Systems of Equations:” “Section 4.2: Systems of Equations, Algebraic Procedures”
Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: Chapter 4: Systems of Equations:” “Section 4.2: Systems of Equations, Algebraic Procedures” (PDF)
Instructions: Please click on the link above, and read Section 4.2 on pages 61–72. Section 4.2 will explore how to find solution(s) for a system of linear equations using elementary operations, Gaussian elimination, and other algebraic procedures. This reading should take you approximately 1 hour and 30 minutes to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.See a broken link? Please let us know!
- Assessment: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 4: Systems of Equations:” “Section 4.3: Exercises”
Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 4: Systems of Equations:” “Section 4.3: Exercises” (PDF)
Instructions: Please click on the link above to open the PDF. Scroll down to page 72, and complete problems 1, 6, 12, 13, 17, 19, 22, 30, 35, and 36. Next, click on “Solutions” (PDF) and check your answers on pages 14–20. This assessment should take you approximately 3 hours and 30 minutes to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.See a broken link? Please let us know!
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 4: Systems of Equations:” “Section 4.2: Systems of Equations, Algebraic Procedures”
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1.4 Matrices
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 5: Matrices:” “Section 5.1: Matrix Arithmetic”
Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 5: Matrices:” “Section 5.1: Matrix Arithmetic” (PDF)
Instructions: Please click on the link above, and read Section 5.1 on pages 77–91. Section 5.1 will provide an overview of matrices and matrix arithmetic. This reading should take you approximately 3 hours to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.See a broken link? Please let us know!
- Assessment: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 5: Matrices:” “Section 5.2: Exercises”
Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 5: Matrices:” “Section 5.2: Exercises” (PDF)
Instructions: Please click on the link above to open the PDF. Scroll down to page 92, and complete problems 3, 5, 6, 7, 10, 12, 16, 19, 24, 28, 32 37, 40, 44, 47, and 53. Next, click on “Solutions” (PDF) and check your answers on pages 19–28. This assessment should take you approximately 5 hours to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.See a broken link? Please let us know!
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 5: Matrices:” “Section 5.1: Matrix Arithmetic”
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Unit 2: Determinants, Rank, and Linear Transformations
Despite the complicated definition of determinants, they are very useful because they allow you to determine, using only one number, whether or not a matrix is invertible. They also are helpful for computing eigenvalues, which you will learn about later. In this unit, you will learn about ways to find the determinants and to use determinants to compute the inverse of a matrix.
Time Advisory show close
You will learn about the rank of the matrix, a very important concept in linear algebra. You will begin by studying the row-reduced echelon form of a matrix and proving that the row-reduced echelon form for a given matrix is unique. This is useful, because you can logically deduce important conclusions about the original matrix by examining its unique row-reduced echelon form. You will then learn that the rank of a matrix is related to the number of linearly independent columns or rows of that matrix; it describes the dimensionality of the space. It is also very important in the use of matrices to solve a system of linear equations, because it tells you whether Ax = 0 has zero, one, or an infinite number of solutions.
Finally, you will learn how matrices also arise in geometry, especially while studying certain types of linear transformations. You will learn that a mxn matrix can be used to transform vectors in Fn to vectors in Fm via matrix multiplication. As you will see, these types of transformation arise quite naturally in linear algebra and are important for applications in mathematics, physics, and engineering.
Learning Outcomes show close
- 2.1 Determinants
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2.1.1 Basic Techniques and Properties
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 6: Determinants:” “Section 6.1: Basic Techniques and Properties”
Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 6: Determinants:” “Section 6.1: Basic Techniques and Properties” (PDF)
Instructions: Please click on the link above, and read Section 6.1 on pages 97–104. Section 6.1 will provide an introduction to determinants and techniques for finding them. This reading should take you approximately 1 hour to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.See a broken link? Please let us know!
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 6: Determinants:” “Section 6.1: Basic Techniques and Properties”
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2.1.2 Applications
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 6: Determinants:” “Section 6.2: Applications”
Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 6: Determinants:” “Section 6.2: Applications” (PDF)
Instructions: Please click on the link above, and read Section 6.2 on pages 104–109. Section 6.2 will introduce some applications, including Cramer’s rule. This reading should take you approximately 1 hour to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages aboveSee a broken link? Please let us know!
- Assessment: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 6: Determinants:” “Section 6.3: Exercises”
Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 6: Determinants:” “Section 6.3: Exercises” (PDF)
Instructions: Please click on the link above to open the PDF. Scroll down to page 109, and complete problems 1, 3, 4, 5, 9, 11, 13, 16, 20, 21, 24, 26, 28, 32, and 36. Next, click on “Solutions” (PDF) and check your answers on pages 28–38. This assessment should take you approximately 5 hours to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.See a broken link? Please let us know!
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 6: Determinants:” “Section 6.2: Applications”
- 2.2 Rank of a Matrix
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2.2.1 Elementary Matrices
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 8: Rank of a Matrix:” “Section 8.1: Elementary Matrices”
Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 8: Rank of a Matrix:” “Section 8.1: Elementary Matrices” (PDF)
Instructions: Please click on the link above, and read Section 8.1 on pages 129–134. Section 8.1 will introduce the elementary matrices, which result from doing a row operation to the identity matrix. This reading should take you approximately 1 hour to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.See a broken link? Please let us know!
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 8: Rank of a Matrix:” “Section 8.1: Elementary Matrices”
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2.2.2 The Row Reduced Echelon Form of a Matrix
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 8: Rank of a Matrix:” “Section 8.2: The Row Reduced Echelon Form of a Matrix”
Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 8: Rank of a Matrix:” “Section 8.2: The Row Reduced Echelon Form of a Matrix” (PDF)
Instructions: Please click on the link above, and read Section 8.2 on pages 135–139. Section 8.2 will review the description of the row-reduced echelon form. This reading should take you approximately 1 hour to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.See a broken link? Please let us know!
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 8: Rank of a Matrix:” “Section 8.2: The Row Reduced Echelon Form of a Matrix”
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2.2.3 The Rank of a Matrix
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 8: Rank of a Matrix:” “Section 8.3: The Rank of a Matrix”
Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 8: Rank of a Matrix:” “Section 8.3: The Rank of a Matrix” (PDF)
Instructions: Please click on the link above, and read Section 8.3 on pages 139–142. Section 8.3 will define rank and explain how to find the rank. This reading should take you approximately 1 hour to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.See a broken link? Please let us know!
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 8: Rank of a Matrix:” “Section 8.3: The Rank of a Matrix”
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2.2.4 Linear Independence and Bases
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 8: Rank of a Matrix:” “Section 8.4: Linear Independence and Bases”
Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 8: Rank of a Matrix:” “Section 8.4: Linear Independence and Bases” (PDF)
Instructions: Please click on the link above, and read Section 8.4 on pages 142–152. Section 8.4 will introduce linear independence and bases. This reading should take you approximately 2 hours to complete.
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- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 8: Rank of a Matrix:” “Section 8.4: Linear Independence and Bases”
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2.2.5 Fredholm Alternative
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 8: Rank of a Matrix:” “Section 8.5: Fredholm Alternative”
Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 8: Rank of a Matrix:” “Section 8.5: Fredholm Alternative” (PDF)
Instructions: Please click on the link above, and read Section 8.5 on pages 153–156. Section 8.5will introduce the Fredholm Alternative for the case of real matrices here. This reading should take you approximately 2 hours to complete.
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- Assessment: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 8: Rank of a Matrix:” “Section 8.6: Exercises”
Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 8: Rank of a Matrix:” “Section 8.6: Exercises” (PDF)
Instructions: Please click on the link above to open the PDF. Scroll down to page 156, and complete problems 2, 5, 7, 10, 12, 16, 18, 25, 32, 34, 45, 50, and 54. Next, click on “Solutions” (PDF) and check your answers on pages 38–49. This assessment should take you approximately 4 hours to complete.
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- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 8: Rank of a Matrix:” “Section 8.5: Fredholm Alternative”
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2.4 Linear Transformations
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 9: Linear Transformations:” “Section 9.1: Linear Transformations” and “Section 9.2: Constructing the Matrix of a Linear Transformation”
Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 9: Linear Transformations:” “Section 9.1: Linear Transformations” and “Section 9.2: Constructing the Matrix of a Linear Transformation” (PDF)
Instructions: Please click on the link above, and read Sections 9.1 and 9.2 on pages 163–173. Section 9.1 and Section 9.2 will introduce linear transformations. These readings should take you approximately 3 hours to complete.
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- Assessment: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 9: Linear Transformations:” “Section 9.3: Exercises”
Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 9: Linear Transformations:” “Section 9.3: Exercises” (PDF)
Instructions: Please click on the above link to open the PDF. Scroll down to page 173, and work on problems 2, 7, 9, 14, 17, 20, 23, 28, 32, 47, and 51. Next, click on “Solutions” (PDF) and check your answers on pages 49–59. This assessment should take you approximately 4 hours to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.See a broken link? Please let us know!
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 9: Linear Transformations:” “Section 9.1: Linear Transformations” and “Section 9.2: Constructing the Matrix of a Linear Transformation”
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Unit 3: Spectral Theory, Matrices, and Inner Product
We have seen that matrix A corresponds to a linear transformation, T, i.e. T(x) =Ax. If the matrix is square, then those nonzero vectors, any of which are transformed to a multiple of themselves, are called eigenvectors of the matrix, and the mutliples by which they are transformed are called eigenvalues (eigenis the German word for characteristic). The set of eigenvalues is called the spectrum of A and plays an important role in linear algebra. In this unit, you will study the spectrum of a square matrix in detail.
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The first thing that we will see is that the eigenvalues of an nxn matrix correspond to the roots of an nth degree polynomial, called the characteristic polynomial of A. As you know from algebra, roots of a polynomial can have multiplicity larger than 1. For eigenvalues, the multiplicity of a root is called the algebraic multiplicity of the eigenvalue, while the dimension of the set of vectors for which it is an eigenvalue is called the geometric multiplicity of the eigenvalue. If these two multiplicities are the same for all eigenvalues, then you will see that A is similar to a matrix with nonzero entries only along the main diagonal, that is A=S-1DS for an invertible matrix S. In fact, the diagonal entries of D are the eigenvalues of A, and the columns of S are a basis of Rn consisting of eigenvectors. Since solving polynomial equations can be difficult, you will study methods of estimating eigenvalues just by looking at the matrix.
You will then consider situations where the eigenvalues are known to be real and the matrix S, which diagonalizes A, can take a special form. If the matrix A is symmetric, that is aij=aji for all i,j, then you will see that A can be diagonalized, that is, all roots of the characteristic polynomial are real numbers. Further, you will learn how to choose the basis of eigenvalues, which comprise S so that the rows and columns each are unit vectors and are mutually perpendicular. Such matrices are called orthogonal and have the property that S-1=ST, where ST is the matrix obtained from S by interchanging its rows and columns. Since not all square matrices are diagonalizable, you will need one of the most important theorems in the spectral theory of matrices: Schur’s Theorem, which is useful for analyzing the structure of matrices.
In linear algebra, you usually want to see whether or not Ax = b has any solutions. Often Ax = b has no solution because there are more equations than unknowns, that is, the linear system is inconsistent and b is not in the column space of A. In this case, you can try to find an approximation using a very important technique of the least square approximation. Another thing you want to do is to characterize when Ax = b has a solution, that is, construct conditions for solvability of the system. One way of doing this is by using Fredholm’s Alternative, which is discussed in this unit. Fredholm’s Alternative is important, because it can be generalized to more general vector spaces, where the concept of rank of a determinant is not defined. Finally, this unit discusses an important result known as the singular value decomposition, which gives a factorization of a matrix.
Learning Outcomes show close
- 3.1 Spectral Theory
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3.1.1 Eigenvalues and Eigenvectors of a Matrix
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 12: Spectral Theory:” “Section 12.1: Eigenvalues and Eigenvectors of a Matrix”
Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 12: Spectral Theory:” “Section 12.1: Eigenvalues and Eigenvectors of a Matrix” (PDF)
Instructions: Please click on the link above, and read Section 12.1 on pages 215–231. Spectral Theory refers to the study of eigenvalues and eigenvectors of a matrix, which is introduced in Section 12.1. This reading should take you approximately 2 hours to complete.
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- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 12: Spectral Theory:” “Section 12.1: Eigenvalues and Eigenvectors of a Matrix”
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3.1.2 The Estimation of Eigenvalues
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 12: Spectral Theory:” “Section 12.3: The Estimation of Eigenvalues”
Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 12: Spectral Theory:” “Section 12.3: The Estimation of Eigenvalues” (PDF)
Instructions: Please click on the link above and read Section 12.3 on pages 236–237. Section 12.3 introduces Gerschgorin’s Theorem, which provides a way to estimate where the eigenvalues are just from looking at the matrix. This reading should take you approximately 30 minutes to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.See a broken link? Please let us know!
- Assessment: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 12: Spectral Theory:” “Section 12.4: Exercises”
Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 12: Spectral Theory:” “Section 12.4: Exercises” (PDF)
Instructions: Please click on the above link to open the PDF. Scroll down to page 237, and complete problems 3, 5, 8, 11, 13, 20, 25, 28, 43, 48, and 54. Next, click on “Solutions” (PDF) and check your answers on pages 76–84. This assessment should take you approximately 4 hours and 30 minutes to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.See a broken link? Please let us know!
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 12: Spectral Theory:” “Section 12.3: The Estimation of Eigenvalues”
- 3.2 Matrices and Inner Product
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3.2.1 Symmetric and Orthogonal Matrices
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 13: Matrices and the Inner Product:” “Section 13.1: Symmetric and Orthogonal Matrices”
Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 13: Matrices and the Inner Product:” “Section 13.1: Symmetric and Orthogonal Matrices” (PDF)
Instructions: Please click on the link above, and read Section 13.1 on pages 245–255. Section 13.1 will introduce symmetric and orthogonal matrices. This reading should take you approximately 2 hours to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.See a broken link? Please let us know!
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 13: Matrices and the Inner Product:” “Section 13.1: Symmetric and Orthogonal Matrices”
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3.2.2 Fundamental Theory and Generalizations
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 13: Matrices and the Inner Product:” “Section 13.2: Fundamental Theory and Generalizations”
Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 13: Matrices and the Inner Product:” “Section 13.2: Fundamental Theory and Generalizations” (PDF)
Instructions: Please click on the link above, and read Section 13.2 on pages 255–262. Sections 13.2 will discuss several results including the Gram-Schmidt process and the Schur’s Theorem. This reading should take you approximately 2 hours to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.See a broken link? Please let us know!
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 13: Matrices and the Inner Product:” “Section 13.2: Fundamental Theory and Generalizations”
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3.2.3 Least Square Approximation
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 13: Matrices and the Inner Product:” “Section 13.3: Least Square Approximation”
Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 13: Matrices and the Inner Product:” “Section 13.3: Least Square Approximation” (PDF)
Instructions: Please click on the link above, and read Section 13.3 on pages 263–266. Sections 13.3 discusses a very important technique known as the Least Square Approximation. This reading should take you approximately 1 hour to complete.
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- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 13: Matrices and the Inner Product:” “Section 13.3: Least Square Approximation”
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3.2.4 Additional Results
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 13: Matrices and the Inner Product:” “Section 13.4: The Right Polar Factorization,” “Section 13.5: The Singular Value Decomposition,” “Section 13.6: Approximation in the Frobenius Norm,” and “Section 13.7: Moore Penrose Inverse”
Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 13: Matrices and the Inner Product:” “Section 13.4: The Right Polar Factorization,” “Section 13.5: The Singular Value Decomposition,” “Section 13.6: Approximation in the Frobenius Norm,” and “Section 13.7: Moore Penrose Inverse” (PDF)
Instructions: Please click on the link above, and read the indicated sections on pages 267–274. Sections 13.4–13.7 will introduce several important topics and results. Please work through these sections carefully. These readings should take you approximately 2 hours to complete.
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- Assessment: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 13: Matrices and the Inner Product:” “Section 13.8: Exercises”
Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 13: Matrices and the Inner Products:” “Section 13.8: Exercises” (PDF)
Instructions: Please click on the link above to open the PDF. Scroll down to page 274, and work on problems 2, 5, 10, 14, 19, 24, 28, 32, 37, 39, 43, 48, 52, and 57. Next, click on “Solutions” (PDF) and check your answers on pages 88–106. This assessment should take you approximately 4 hours to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.See a broken link? Please let us know!
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 13: Matrices and the Inner Product:” “Section 13.4: The Right Polar Factorization,” “Section 13.5: The Singular Value Decomposition,” “Section 13.6: Approximation in the Frobenius Norm,” and “Section 13.7: Moore Penrose Inverse”
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Unit 4: Vector Spaces and Linear Transformations
This unit discusses vector spaces and linear transformations. The first topic you will study in this unit is that of an abstract vector space. You will see that a vector space is a collection of vectors that satisfies a set of axioms. Next, you will study ideas such as subspaces, linear independence, and bases in the context of vector spaces. Remember to think of R or C if you are confused. Next, you will construct abstract fields and vector spaces. You will begin this by first reviewing some basic algebra relating to polynomials. This is both interesting and important, because it provides the basis for constructing different kinds of fields. Finally, you will look at inner product spaces, which are vector spaces that also have an inner product, before moving on to linear transformations, which is the second topic of study in this unit.
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Linear transformations have many applications within mathematics as well as in fields outside of mathematics, such as physics. You will study the basic definitions of linear transformations and properties and relations between these and matrices.
Learning Outcomes show close
- 4.1 Vector Spaces
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4.1.1 Algebraic Considerations, Linear Independence, and Bases
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 16: Vector Spaces:” “Section 16.1: Algebraic Considerations” and “Section 16.2.1: Linear Independence and Bases”
Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 16: Vector Spaces:” “Section 16.1: Algebraic Considerations” and “Section 16.2.1: Linear Independence and Bases” (PDF)
Instructions: Please click on the link above, and read Section 16.1 on page 323 and section 16.2.1 on pages 325–330. Section 16.1 will provide the definition of a vector space, and Section 16.2.1 will discuss subspaces, linear dependence, and bases. These readings should take you approximately 2 hours to complete.
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- Assessment: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 16: Vector Spaces:” “Section 16.2: Exercises”
Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 16: Vector Spaces:” “Section 16.2: Exercises” (PDF)
Instructions: Please click on the link above to open the PDF. Scroll down to page 330, and complete problems 1, 2, 3, and 4. Next, click on “Solutions” (PDF) and check your answers on pages 124–125. This assessment should take you approximately 2 hours to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.See a broken link? Please let us know!
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 16: Vector Spaces:” “Section 16.1: Algebraic Considerations” and “Section 16.2.1: Linear Independence and Bases”
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4.1.2 Vector Spaces and Fields
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 16: Vector Spaces:” “Section 16.3: Vector Space and Fields”
Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 16: Vector Spaces:” “Section 16.3: Vector Space and Fields” (PDF)
Instructions: Please click on the link above, and read Section 16.3 on pages 330–343. Section 16.3 will discuss vector space and fields. This reading should take you approximately 3 hours and 30 minutes to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.See a broken link? Please let us know!
- Assessment: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 16: Vector Spaces:” “Section 16.4: Exercises”
Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 16: Vector Spaces:” “Section 16.4: Exercises” (PDF)
Instructions: Please click on the link above to open the PDF. Scroll down to page 243, and work on problems 2, 4, 10, 12, 16, 20, 24, and 29. Next, click on “Solutions” (PDF) and check your answers on pages 125–135. This assessment should take you approximately 4 hours to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.See a broken link? Please let us know!
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 16: Vector Spaces:” “Section 16.3: Vector Space and Fields”
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4.1.3 Inner Product Spaces
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 16: Vector Spaces:” “Section 16.5: Inner Product Spaces”
Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 16: Vector Spaces:” “Section 16.5: Inner Product Spaces” (PDF)
Instructions: Please click on the link above, and read Section 16.5 on pages 348–361. Section 16.5 will discuss vector spaces with inner products. This reading should take you approximately 3 hours and 30 minutes to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.See a broken link? Please let us know!
- Assessment: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 16: Vector Spaces:” “Section 16.6: Exercises”
Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 16: Vector Spaces:” “Section 16.6: Exercises” (PDF)
Instructions: Please click on the above link to open the PDF. Scroll down to page 361, and complete problems 1, 4, 7, 12, 15, 17, and 22. Next, click on “Solutions” (PDF) and check your answers on pages 135–150. This assessment should take you approximately 4 hours to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.See a broken link? Please let us know!
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 16: Vector Spaces:” “Section 16.5: Inner Product Spaces”
- 4.2 Linear Transformations
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4.2.1 Matrix Multiplication and L(V,W)
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 17: Linear Transformations:” “Section 17.1: Matrix Multiplication as a Linear Transformation” and “Section 17.2: L(V,W) as a Vector Space”
Link: Professor Kenneth Kuttler’s Elementary Linear Algebra:“Chapter 17: Linear Transformations:” “Section 17.1: Matrix Multiplication as a Linear Transformation” and “Section 17.2: L(V,W) as a Vector Space” (PDF)
Instructions: Please click on the link above, and read the indicated sections on pages 367 and 368. Sections 17.1 and 17.2 will discuss linear transformations and vector spaces. These readings should take you approximately 30 minutes to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.See a broken link? Please let us know!
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 17: Linear Transformations:” “Section 17.1: Matrix Multiplication as a Linear Transformation” and “Section 17.2: L(V,W) as a Vector Space”
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4.2.2 Eigenvalues And Eigenvectors Of Linear Transformations
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 17: Linear Transformations:” “Section 17.3: Eigenvalues and Eigenvectors of Linear Transformations”
Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 17: Linear Transformations:” “Section 17.3: Eigenvalues and Eigenvectors of Linear Transformations” (PDF)
Instructions: Please click on the above link, and read Section 17.3 on pages 369–373. Section 17.3 will discuss finding eigenvalues and eigenvectors of linear transformations. This reading should take you approximately 1 hour and 30 minutes to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.See a broken link? Please let us know!
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 17: Linear Transformations:” “Section 17.3: Eigenvalues and Eigenvectors of Linear Transformations”
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4.2.3 The Block Diagonal Matrices
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 17: Linear Transformations:” “Section 17.4: Block Diagonal Matrices”
Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 17: Linear Transformations:” “Section 17.4: Block Diagonal Matrices” (PDF)
Instructions: Please click on the link above, and read Section 17.4 on pages 373–377. Sections 17.4 will discuss linear transformations, which will result by multiplication by n × n matrices. This reading should take you approximately 1 hour to complete.
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- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 17: Linear Transformations:” “Section 17.4: Block Diagonal Matrices”
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4.2.4 The Matrix of a Linear Transformation
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 17: Linear Transformations:” “Section 17.5: The Matrix of a Linear Transformation”
Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 17: Linear Transformations:” “Section 17.5: The Matrix of a Linear Transformatio” (PDF)
Instructions: Please click on the link above, and read Section 17.5 on pages 377–386. Section 17.4 will discuss matrices of linear transformations. This reading should take you approximately 2 hours to complete.
Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.See a broken link? Please let us know!
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 17: Linear Transformations:” “Section 17.5: The Matrix of a Linear Transformation”
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4.2.5 The Matrix Exponential, Differential Equations
- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 17: Linear Transformations:” “Section 17.6: The Matrix Exponential, Differential Equations”
Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 17: Linear Transformations:” “Section 17.6: The Matrix Exponential, Differential Equations” (PDF)
Instructions: Please click on the link above, and read Section 17.6 on pages 386–391. Section 17.6 will introduce fundamental matrices, matrices whose entries are differentiable functions. This reading should take you approximately 1 hour and 30 minutes to complete.
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- Assessment: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 17: Linear Transformations:” “Section 17.7: Exercises”
Link: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 17: Linear Transformations:” “Section 17.7: Exercises” (PDF)
Instructions: Please click on the link above to open the PDF. Scroll down to page 391, and work on problems 1, 3, 8, 12, 15, 18, and 21. Next, click on “Solutions” (PDF) and check your answers on pages 151–157. This assessment should take you approximately 4 hours to complete.
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- Reading: Professor Kenneth Kuttler’s Elementary Linear Algebra: “Chapter 17: Linear Transformations:” “Section 17.6: The Matrix Exponential, Differential Equations”
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Final Exam
- Final Exam: The Saylor Foundation's "MA211 Final Exam"
Link: The Saylor Foundation's "MA211 Final Exam" (HTML)
Instructions: You must be logged into your Saylor Foundation School account in order to access this exam. If you do not yet have an account, you will be able to create one, free of charge, after clicking the link.See a broken link? Please let us know!
- Final Exam: The Saylor Foundation's "MA211 Final Exam"
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