Course Outline for Mathematics 6
Elementary Linear Algebra

Effective: Fall 2018
SLO Rev: 12/06/2016

Catalog Description:

MTH 6 - Elementary Linear Algebra

3.00 Units

Introduction to linear algebra: matrices, determinants, systems of equations, vector spaces, linear transformations, eigenvalue, eigenvectors, and applications.
Prerequisite: MTH 2.
CB03: TOP Code 1701.00 - Mathematics, General
Course Grading: Letter Grade Only
Type Units Inside of Class Hours Outside of Class Hours Total Student Learning Hours
Lecture 3.00 54.00 108.00 162.00
Laboratory 0.00 18.00 0.00 18.00
Total 3.00 72.00 108.00 180.00

Measurable Objectives:

Upon completion of this course, the student should be able to:
  1. solve systems of linear equations using Gaussian elimination, Gauss-Jordan elimination, inverse matrices, row operations and determinants;
  2. compute determinants of all orders and apply the properties of determinants;
  3. perform algebraic operations on matrices and be able to construct their inverses, adjoints, transposes;
  4. determine the rank of a matrix and relate this to systems of equations;
  5. perform vector algebra in R^n;
  6. recognize and use the properties of vector spaces and inner product spaces;
  7. discuss the relationship between the dimensions of the null space, the rank and the number of columns of a matrix;
  8. prove a given set of vectors of R^n is a subspace of R^n;
  9. prove if a given set of vectors are linearly independent and spans a given subspace;
  10. determine the dimension of a subspace and produce a basis for the subspace;
  11. use the Gram-Schmidt process to create an orthonormal basis for a subspace;
  12. use bases and orthonormal bases to solve problems;
  13. find the column space, row space and null space of a matrix;
  14. discuss the concepts of subspaces, linear independence, bases, orthogonality, and their relation;
  15. explain the concept of a linear transformation from a vector space V to a vector space W;
  16. identify and use the properties of linear transformations and their relation to matrices;
  17. describe the kernel and the range of a linear transformation;
  18. prove basic properties of eigenvectors and eigenvalues;
  19. diagonalize and orthogonally diagonalize matrices;
  20. use eigenvectors and eigenvalues to solve applications;
  21. prove basic results in linear algebra.

Course Content:

  1. Solution of systems of linear equations and matrices
    1. Gaussian elimination
    2. Gauss-Jordan elimination
    3. Algebra of matrices
    4. Inverse matrices
    5. Relationship between coefficient matrix invertiablility and solutions of the system
    6. Transposes
    7. Determinants and their properties
    8. Special matrices
      1. Diagonal
      2. Triangular
      3. Symmetric
  2. Vector spaces
    1. Algebra of R^n
      1. Orthogonality of two vectors
    2. Subspaces
    3. Subspaces related to a matrix
      1. Row, column, and null space
      2. Rank and nullity
    4. Linear independence and dependence
    5. Basis and dimension of a vector space
      1. Orthonormal basis
      2. Gram-Schmidt process
      3. Change of basis
      4. Rank and dimension
    6. Inner product spaces
      1. Dot product
      2. Norm of the vector
      3. Angle between two vectors
  3. Linear transformations
    1. Properties and matrix representations
    2. Geometry of linear transformation
    3. Kernal and range
    4. Rank and nullity
    5. Inverse linear transformation
    6. Matrices of general linear transformation
  4. Eigenvectors and eigenvalues
    1. Eigenspaces
    2. Diagonalization
    3. Orthogonal diagonalization
    4. Applications
  5. Proofs of basic theorems
  6. Applications:  quadratic forms, the Principal Axes Theorem, approximation, Fourier series (if time permits)

Methods of Instruction:

  1. Lecture/Discussion
  2. Demonstration/Exercise
  3. Problem Solving
  4. Group Activities
  5. Distance Education

Assignments and Methods of Evaluating Student Progress:

1. Typical Assignments
  1. Find the standard matrix for the stated composition of linear operators on R^2. 1) A rotation of 90 degrees, followed by a reflection about the line y = x. 2) An orthogonal projection on the y-axis, followed by a contraction with factor k = 0.5. 3) A reflection about the x-axis, followed by a dilation with factor k = 3.
  2. Let V be an inner product space. Show that if w is orthogonal to both u1 and u2, it is orthogonal to k1u1 + k2u2 for all scalars k1 and k2. Interpret this result geometrically in the case where V is R3 with the Euclidean inner product.
2. Methods of Evaluating Student Progress
  1. Exams/Tests
  2. Quizzes
  3. Homework
  4. Final Examination
  5. Lab Activities
3. Student Learning Outcomes
Upon the completion of this course, the student should be able to:
  1. Critically analyze mathematical problems using a logical methodology.
  2. Communicate mathematical ideas, understand definitions, and interpret concepts.
  3. Increase confidence in understanding mathematical concepts, communicating ideas and thinking analytically.

Textbooks (Typical):

  1. Anton, H. (2014). Elementary Linear Algebra (11 th). Wiley Publishing.
Additional Materials:
  • A calculator may be required.

Abbreviated Class Schedule Description:

Introduction to linear algebra: matrices, determinants, systems of equations, vector spaces, linear transformations, eigenvalue, eigenvectors, and applications.
Prerequisite: MTH 2.

Discipline:
Mathematics*