General information

• General description. Math 130 is a requirement for mathematics and physics majors, and it's highly recommended for majors in other sciences especially including computer-science majors. Topics include systems of linear equations and their solutions, matrices and matrix algebra, inverse matrices; determinants and permutations; real n-dimensional vector spaces, abstract vector spaces and their axioms, linear transformations; inner products (dot products), orthogonality, cross products, and their geometric applications; subspaces, linear independence, bases for vector spaces, dimension, matrix rank; eigenvectors, eigenvalues, matrix diagonalization. Some applications of linear algebra will be discussed, such as computer graphics, Kirchoff's laws, linear regression (least squares), Fourier series, or differential equations.
  See also Clark's Academic Catalog.

• Prerequisites. The prerequisite for the course is one year of college calculus, others by permission only.

• Web pages for related courses
  Math 120, Calculus I
  Math 121, Calculus II
  Math 122, Calculus III
  Math 131, Multivariate Calculus
  Math 217, Probablility and Statistics
  Math 218, Mathamatical Statistics
  Math 225, Modern Algebra

• Course Hours. MWF 11:00–11:50.

• Office hours.

• Course Schedule

• Assignments & tests. There will be numerous short assignments, mostly from the text, occasional quizzes, two tests during the semester, and a two-hour final exam during finals week in December. The two tests during the semester are yet to be scheduled.

• Course grade. The course grade will be determined as follows:
  2/9 assignments and quizzes,
  2/9 each of the two midterms, and
  1/3 for the final exam.

• Course goals.
  • To provide students with a good understanding of the concepts and methods of linear algebra, described in detail in the syllabus.
  • To help the students develop the ability to solve problems using linear algebra.
  • To connect linear algebra to other fields both within and without mathematics.
  • To develop abstract and critical reasoning by studying logical proofs and the axiomatic method as applied to linear algebra.

• Course objectives. Students will be able to apply the concepts and methods described in the syllabus, they will be able to solve problems using linear algebra, they will know a number of applications of linear algebra, and they will be able to follow complex logical arguments and develop modest logical arguments. The text and class discussion will introduce the concepts, methods, applications, and logical arguments; students will practice them and solve problems on daily assignments, and they will be tested on quizzes, midterms, and the final.

• Textbook. The text for the course is Introductory Linear Algebra, an Applied First Course, 8th edition, Bernard Kolman and David R. Hill. Prentice-Hall, 2005. ISBN: 0-13-143740-2. Prentice Hall's Companion Website for the book.

Syllabus

1. Linear equations and matrices. About 6 meetings.

§ 1.1   Linear systems.
            Exercises 1–4, 13, 14, 21, 22, 23, T4.
§ 1.2   Matrices.
            Exercises 1, 2, 4–7 parts a–d each, 8, 9, T1, T5a, ML1, ML2.
§ 1.3   Dot product and matrix multiplication.
            Exercises 1–4, 7, 9, 11, 12, 19, 20, 33, T1, T4, ML1, ML2, ML5, ML6.
§ 1.4   Properties of matrix operations.
            Exercises 11–13, 19, T.10, T.24, T.30, ML1, ML3, ML4, ML7.
§ 1.5   Matrix transformations.
            Exercises 1, 2, 5, 6, 15, 16, 17.
§ 1.6   Solutions of linear systems of equations.
            Exercises 1–8, 13, 14, 20, 28, T8.
§ 1.7   The inverse of a matrix.
            Exercises 1–4, 9, 11, 12, 17a, 22, T4, T5, ML1–ML4.

Topics: systems of simultaneous linear equations; method of elimination (sometimes called Gaussian elimination or Gauss-Jordan reduction); intersections of subspaces (lines and planes); matrices (their rows, columns, and notation for matrices); vectors; addition, subtraction, and multiplication of matrices and of vectors; linear combinations; dot products (also called inner products); matrix multiplication; properties (theory) of matrix operations; transformations in R² and R³ described by matrices including reflections, dilations, contractions, and rotations; solutions of systems of linear equations (that is, details of Gaussian elimination); row echelon form and reduced row echelon form; elementary row operation; consistency and inconsistency of systems; homogeneous systems of equations; invertible square matrices (also called nonsingular matrices); inverses of matrices; algorithm for inverting a matrix.

Comments: This chapter covers the fundamental concepts of the subject, and as such are very important. You need to understand many of them in order to understand the later chapters. Fortunately, a couple of these concepts you're seen before, like how to solve a system of linear equations.

2. Applications. About 3 meetings.

Comments: We'll look at two or three of the applications in chapter 2, depending on class preferences. We probably won't look at them right after discussing chapter 1, but select convenient times during the first half of the course. Here's a list of topics from that chapter.

§ 2.1   Error detection/correction codes. These are used in computer science. One that you may have heard of is a parity bit.
§ 2.2   Graphs. The kind of graph under discussion has a set of vertices (also called nodes) and directed edges between them (also called arrows). They're used in mathematics, social sciences, and computer science.
§ 2.3   Computer graphics. Continues the discussion we started in section 1.5 on transformations of the plane.
§ 2.5   Electrical circuits. Kirchhoff's laws for circuits.
§ 2.6   Markov chains. Statistical models where there are several possible states, and at each time unit, the given state will change according to a probablility matrix to one of the other states.
§ 2.7   Linear economic models. Leontief models and others that describe economic transformations in terms of matrices. Analogous to the Markov chains in 2.5, but not probabilistic.
§ 2.8   Wavelets for data compression. The JPEG format used to store images and transmit them across the internet uses uses this compression scheme.

3. Determinants. About 5 meetings.

§ 3.1   Definition and properties
            Exercises parts a and b of: 1–6, 15, 19, 20, T1, T6, and T12.
§ 3.2   Cofactor expansion and applications
            Exercises 1, 3ab, 4ab, 10ab, 11ab, 20, 23.
§ 3.3   Determinants from a computational point of view

Topics: permutations and inversions of permutations; even and odd permutations; determinants of square matrices; theory of determinants; singular matrices and determinants; cofactors and minors of square matrices; adjoints and inverse matrices; Cramer's rule; efficiency of algorithms to invert matrices.

Comments: You've probably seen determinants before, but not everyone has. We'll be using them throughout the rest of the course. There are many different ways to introduce them, but we'll follow Kolman and Hill's presentation in terms of permutations.

4. Vectors in Rⁿ. About 4 meetings.

§ 4.1   Vectors in the plane
            Exercises 3, 5, 9ab, 13, 19, 21ab, 22ab, 23, 24, 27, 28, T5, T7.
§ 4.2   n-vectors
            Exercises 8, 10ab, 11ab, 12ab, 14, 21ab, 23, 25, 26, 27ab, 31, 32, 34, T7, T10, ML2–ML5, ML8–ML9.
§ 4.3   Linear transformations
            Exercises 1, 4–7, 21, 22, 25, 26, 27, 29, T9, T10, T11.

Topics: vectors (points) in Rⁿ; length (norm) of vectors; unit vectors; determinants and area of triangles and parallelograms; graphical interpretation of vector operations; angle between two vectors and dot products; vectors in higher dimensional space; orientation in space; properties of dot products; the Cauchy-Schwarz inequality; orthogonality of vectors ("orthogonal" is another word for "perpendicular"); unit vectors and standard unit vectors in Rⁿ; linear transformations L: Rⁿ → R^m.

Comments: There are many places we could have started linear algebra. We happened to use simultaneous linear equations in chapter 1. An alternative beginning is with vectors. Vectors allow us to connect linear algebra to geometry.

5. Applications of vectors in R² and R³. About 3 meetings.

§ 5.1   Cross product in R³
            Exercises 1ab, 2ab, 9–12, T2, T3, T4, T5, (T7), ML1–2, ML5–6.
§ 5.2   Lines and planes
            Exercises 5ab, 6ab, 10ab, 13, T3.

Topics: cross product in R³; properties of cross product; cross product and areas of triangles and parallelograms; triple product in R³ and determinants; lines in R² and R³; planes in R³.

Comments: Although this is a short chapter, it's really important. Many of the applications of linear algebra to multivariate calculus and differential geometry are based on the concepts we develop in this chapter.

6. Real vector spaces. About 10 meetings.

§ 6.1   Vector spaces
            Exercises 11, 15, 20, T3.
§ 6.2   Subspaces
            Exercises 1–3, 5, 6, 9, 14, 17–19, 23–25, 27, T4, T10, ML3, ML5.
§ 6.3   Linear independence
            Exercises 1, 2, 6–10, ML1, ML2.
§ 6.4   Basis and dimension
§ 6.5   Homogeneous systems
§ 6.6   The rank of a matrix and applications
§ 6.7   Coordinates and change of basis

Topics: Definition of abstract real vector space; other examples of vector spaces, vector spaces over fields other then R; basic theory of vector spaces; subspaces of vector spaces; linear combinations (again); span of a set of vectors in a vector space, and the geometric interpretation of the span; linear dependence and linear independence of a set of vectors, and their geometric interpretation; basis of a vector space; dimension of a vector space; finite- versus infinite- dimensional vector spaces; null space of a matrix or homogeneous system of equations; nullity; row space of a matrix; row rank of a matrix, rank of a matrix; rank and singularity of a matrix; coordinates in a vector space relative to a given basis; transition matrix between two different bases of a vector space

Comments: Here we treat linear algebra much more abstractly than we have in the preceding chapters. We'll see how the axiomatic method works as a foundation of linear algebra and you'll learn just how to use it to prove important, fundamental theorems, such as the existance of the dimension of a vector space.

8. Eigenvalues, eigenvectors, and diagonalization. About 5 meetings.

§ 8.1   Eigenvalues and eigenvectors
§ 8.2   Diagonalization

Topics: Eigenvalue of a square matrix (also called characteristic value); eigenvectors associated to a given eigenvalue; geometric interpretation of eigenvectors and eigenvalues; algorithm for computing eigenvalues and eigenvectors; characteristic polynomial of a square matrix and its relation to eigenvalues; similar square matrices; similar matrices and eigenvalues; diagonalizable matrices.

Comments: You might think of this as a capstone topic for the course. We connect the geometric concepts of the earlier chapters to the algebraic concepts of eigenvalues and eigenvectors.

Class notes, quizzes, tests, homework assignments

1. Monday, Aug 31. Notes: Welcome.
   Topic: simultaneous systems of linear equations.
   The Chinese method of elimination at http://aleph0.clarku.edu/~djoyce/ma105/simultaneous.html
   MathWorks' MATLAB Tutorial at http://www.mathworks.com/academia/student_center/tutorials/launchpad.html
2. Wednesday, Sep 2. Notes
   Linear systems. Introduction to matrices.
   Asssignment due. §1.1: 1–4, 13, 14, 21, 22, 23, T4.
3. Friday, Sep 4. Notes
   Vectors and their dot products, matrix multiplication, summation notation.
   Asssignment due. §1.2: 1, 2, 4–7 parts a–d each, 8, 9, T1, T5a.
   
   Monday, Sep 7. No class. Martin Luther King, Jr. Day
   
   
4. Wednesday, Sep 9. Notes
   Properties of the matrix operations, powers of matrices, symmetric matrices.
   Asssignment due. §1.3: 1–4, 7, 9, 11, 12, 19, 20, 33, T1, T4.
5. Friday, Sep 11. Continue discussion of properties of matrix operations. Notes
   Quiz covering through §1.3. Answers.
6. Monday, Sep 14. Matrix transformations. Linear transformations of the plane, including rotations, reflections, contractions and expansions. Linear transformations of 3-space. See the notes from Friday.
   Asssignment due. §1.4: 11, 12, 13, 19, T.10, T.24, and T.30.
7. Wednesday, Sep 16. Elemetary row operations, row echelon form, reduced ro echelon form; homogeneous systems. Notes
8. Friday, Sep 18. Matlab. Notes
   Learning to Love Linux
9. Monday, Sep 21.Inverse matrices, uniqueness of the inverse, properties of inverses. How to find the inverse of a matrix. Notes
   Asssignment due. §1.5: 1, 2, 5, 6, 15, 16, 17.
10. Wednesday, Sep 23. A proof that the method described to find a matrix inverse works. Notes
    Asssignment due. §1.6: 1–8, 13, 14, 20, 28, T8.
11. Friday, Sep 25. Introduction to determinants. Determinants of small matrices, permutations, permutation matrices, definition of the determinant in terms of permutations. Notes
    Asssignment due. §1.7: 1–4, 9, 11, 12, 17a, 22, T4, T5, ML1–ML4.
12. Monday, Sep 28. Properties of determinants: transposition, effect of elementary row operations, multilinearity. Notes
13. Wednesday, Sep 30. More properties of determinants: determinants of diagonal and triangular matrices, row reduction to compute matrices, product of matrices. Notes
    Asssignment due. §3.1: parts a and b of: 1–6, 15, 19, and 20. Answers
14. Friday, Oct 2.
    Asssignment due. §3.1: T1, T6, and T12. Answers
15. Monday, Oct 5. First test on chapter 1 and section 3.1. Test, answers.
16. Wednesday, Oct 7. Cofactor expansion, adjoints of matrices, inverse matrices, Cramer's rule.
17. Friday, Oct 9. Vectors in the plane, length of a vector, geometric interpretation of vector addition, properties of dot products of vectors, relation between the dot product of vectors and the cosine of the angle between them, orthogonal vectors. Notes
    Asssignment due. §3.2: 1, 3ab, 4ab, 10ab, 11ab, 20, 23. Answers
    
    Monday, Oct 12. No class. Fall break.
    
    
18. Wednesday, Oct 14. Areas of triangles and parallelograms, unit vectors, vectors in dimension n, and coordinates for physical 3-space Notes
    Asssignment due. §4.1: 3, 5, 9ab, 13, 19, 21ab, 22ab.
19. Friday, Oct 16. Length of a vector, dot products, and angles, all in n-space, the triangle and Cauchy inequalities. Notes
    Asssignment due. §4.1: 23, 24, 27, 28, T5, T7. Answers
20. Monday, Oct 19. Unit vectors and standard unit vectors in Rⁿ; linear transformations L: Rⁿ → R^m. Notes
    Assignment due. §4.2: 8, 10ab, 11ab, 12ab, 14, 21ab, 23, ML2–ML5, ML8–ML9. Answers
21. Wednesday, Oct 21. Linear transformations L: Rⁿ → R^m correspond to m by n matrices. Introduction to graphs and their adjacency matrices. Notes
    Assignment due. §4.2: 25, 26, 27ab, 31, 32, 34, T7, T10. Answers
22. Friday, Oct 23. Cross products in R³: their definition, properties, and length; and triple scalar products. Notes
    Quiz. on sections 4.1 and 4.2. Answers.
    Notes on Quaternions.
23. Monday, Oct 26. The cross product and triangles, parallelograms, and parallelepipeds. Notes
    Assignment due. §4.3: 1, 4–7, 21, 22. Answers
24. Wednesday, Oct 28. Vector equations for lines in the plane and planes in space Notes
    Assignment due. §4.3: 25, 26, 27, 29, T9, T10, T11 Answers
25. Friday, Oct 30. A determinant for describing the line in the plane determined by two points, and for a plane in space determined by three points; parametric descriptions of lines and planes. Introduce abstract vector spaces; precise definition and several examples. Notes
    Assignment due. §5.1: 1ab, 2ab, 9–12, T2, T3, T4, T5, (T7), ML1–2, ML5–6. Answers
26. Monday, Nov 2. Properties of abstract vector spaces. Subspaces: definition and characterizations; subspaces of R². Notes
    Assignment due. §5.2: 5ab, 6ab, 10ab, 13, T3. Answers
27. Wednesday, Nov 4. Subspaces of and R³, solution spaces, spans. Notes
    Kinds of proofs
    Assignment due. §6.1: 11, 15, 20, T3. Answers
28. Friday, Nov 6. Isomorphism, duality; spanning sets and linear independence. Notes
    Assignment due. §6.2: 1–3, 5, 6, 9, 14, 17–19. Answers
29. Monday, Nov 9. Basis of a vector space. (See previous notes.)
    Assignment due. §6.2: 23–25, 27, T4, T10, ML3, ML5. Answers
30. Wednesday, Nov 11. Dimension of a vector space. Notes
    Assignment due. §6.3: 1, 2, 6–10, ML1, ML2. Answers
31. Friday, Nov 13. Second test. Covers 3.2, 4.1–4.3, 5.1–5.2, and 6.1–6.2.
32. Monday, Nov 16. Return test. A constructive method to select linearly independent vectors from a spanning set. Notes
33. Wednesday, Nov 18. Homogeneous and nonhomomogeneous systems of linear equations; introduction to row and column spaces of a matrix, row rank and column rank of a matrix. Notes
    Assignment due. §6.4: 1, 2, 7, 9, 11, 15, 18, 20. Answers
34. Friday, Nov 20. The row rank and the column rank of a matrix are the same; rank and nullity of a matrix; coordinates with respect to a basis of a vector space. Notes
    Assignment due. § 6.4: 22, 28, 31, 35, T2, ML1, ML3, ML7.
35. Monday, Nov 23.
    Assignment due. § 6.5: 1, 3, 4, 8, ML1--ML3.
    
    Thanksgiving break.
    
    
36. Monday, Nov 30.
    Assignment due. § 6.6: 3–6, 18–22, T6, T10, ML1, ML4.

Recent midterms and final

• First midterm, answers
• Second midterm, answers
• Final examination, answers

Pages on the web that you may find interesting

• Humor in mathematics
• Dave's Short Trig Course, and Dave's Short Course on Complex Numbers.
• MathArchives links for Linear and Matrix Algebra.
• Articles by J J O'Connor and E F Robertson on the history of Abstract linear spaces and on the history of Matrices and determinants at The MacTutor History of Mathematics archive.
• Dave Rusin's The Mathematical Atlas describes the subject of linear algebra and its subfields, at Northern Illinois University.
• On-line textbooks:
  • Linear Algebra: An Introduction to Linear Algebra for Pre-Calculus Students. by Tamara Anthony Carter, Richard A. Tapia, and Anne Papakonstantinou, at Rice University. You may find this text useful especially for the first part of our course.
  • Linear Algebra by Jim Hefferon, at Saint Michael's College. This text has many of topics in our syllabus.
  • Elementary Linear Algebra, lecture notes by Keith Matthews. This text also has many of the topics in our syllabus.
  • Linear Algebra and Applications Textbook by Thomas S. Shores. Again, many of the topics in our syllabus.

This page is located on the web at

http://aleph0.clarku.edu/~djoyce/ma130/

David E. Joyce,