Active Outline

General Information

Course ID (CB01A and CB01B)
MATHD02BH
Course Title (CB02)
Linear Algebra - HONORS
Effective Term
Fall 2023
Course Description
Linear algebra and selected topics of mathematical analysis. As an honors course the students will be expected to complete extra assignments to gain deeper insight into linear algebra.
Course Family
Not Applicable

Course Justification

This course satisfies the CSU GE/Breadth requirement Area B4 Mathematics and Quantitative Reasoning. This course satisfies the IGETC requirement Area 2 Mathematical Concepts and Quantitative Reasoning. This course satisfies the mathematics proficiency requirement for an AA degree and is a required core course for the AS-T degree in Mathematics. This is the second course in a sequence of two courses beyond the calculus sequence. This course emphasizes concepts in linear algebra. The content in this course is required for advanced courses in mathematics and the sciences. This course is the honors version of linear algebra and as a result includes more advanced assignments and assessments.

Foothill Equivalency

Does the course have a Foothill equivalent?
Yes
Foothill Course ID
MATH F002B

Course Philosophy

Formerly Statement

Course Development Options

Basic Skill Status (CB08)
Course is not a basic skills course.
Grade Options
  • Letter Grade
  • Pass/No Pass
Repeat Limit
0

Transferability & Gen. Ed. Options

Transferability
Transferable to both UC and CSU
CSU GEArea(s)StatusDetails
CGB4CSU GE Area B4 - Mathematics/Quantitative ReasoningApproved
IGETCArea(s)StatusDetails
IG2XIGETC Area 2 - Mathematical Concepts and Quantitative ReasoningApproved
C-IDArea(s)StatusDetails
MATHMathematicsApprovedC-ID MATH 250

Units and Hours

Summary

Minimum Credit Units
5.0
Maximum Credit Units
5.0

Weekly Student Hours

TypeIn ClassOut of Class
Lecture Hours5.010.0
Laboratory Hours0.00.0

Course Student Hours

Course Duration (Weeks)
12.0
Hours per unit divisor
36.0
Course In-Class (Contact) Hours
Lecture
60.0
Laboratory
0.0
Total
60.0
Course Out-of-Class Hours
Lecture
120.0
Laboratory
0.0
NA
0.0
Total
120.0

Prerequisite(s)

MATH D001D or MATH D01DH (with a grade of C or better)

Corequisite(s)

Advisory(ies)

ESL D272. and ESL D273., or ESL D472. and ESL D473., or eligibility for EWRT D001A or EWRT D01AH or ESL D005.

Limitation(s) on Enrollment

  • (Not open to students with credit in the non-Honors related course.)
  • (Admission into this course requires consent of the Honors Program Coordinator.)

Entrance Skill(s)

General Course Statement(s)

(See general education pages for the requirements this course meets.)

Methods of Instruction

Lecture and visual aids

Discussion of assigned reading

Discussion and problem solving as a class activity

Collaborative learning and small group exercises

Collaborative projects

Use of various technologies including graphing utilities and computer labs

Quiz and examination review performed in class

Homework and extended projects

Guest speakers

Problem solving and exploration activities using applications software

Problem solving and exploration activities using courseware

Assignments

  1. Required readings from text
  2. Problem-solving exercises some including technology
  3. A selection of homework/quizzes, group projects, exploratory worksheets.
  4. Optional project synthesizing various concepts and skills from course content
  5. In addition, the honors project assignment should include completion of additional sets of advanced problems that require a deeper understanding of the topics and/or a written research report (10 to 15 pages).

Methods of Evaluation

  1. Periodic quizzes and/or assignments from sources related to the topics listed in the curriculum are evaluated for completion and accuracy in order to assess student’s comprehension and ability to communicate orally or in writing of course content.
  2. Projects (optional)

    Projects may be used to enhance the student's understanding of topics studied in the course in group or individual formats where communicating their understanding orally through classroom presentation or in writing. The evaluation to be based comprehension of course content.
  3. At least three one-hour exams without projects, or at least two one-hour exams with projects are required. In these evaluations the student is expected to provide complete and accurate solutions to problems that include both theory and application by integrating methods and techniques studied in the course.
  4. A final examination in which the student is expected to display comprehension of course content and be able to choose methods and techniques appropriate to the various problems covered by content in the course outline.
  5. The honors project will be evaluated based on depth of understanding and mastery of advanced techniques employed within the project.

Essential Student Materials/Essential College Facilities

Essential Student Materials: 
  • None.
Essential College Facilities:
  • None.

Examples of Primary Texts and References

AuthorTitlePublisherDate/EditionISBN
* Anton, Howard. "Elementary Linear Algebra, Applications Version", 11th edition, 2014, John Wiley
David C. Lay, "Linear Algebra And Its Applications", 5th Edition, Addison Wesley Publisher, 2015.
Larson, Edwards and Flavo, "Elementary Linear Algebra", 8th Edition, Houghton Mifflin Publisher, 2017.

Examples of Supporting Texts and References

AuthorTitlePublisher
Anton, Howard. "Elementary Linear Algebra", 11th edition, New York, NY: John Wiley and Sons, Inc., 2014
Foley. James D. "Introduction to Computer Graphics", Addison-Wesley, 3rd edition (Supplement for computer graphics applications)
Goodaire, Edgar G. "Linear Algebra: A Pure and Applied First Course", Prentice Hall, 2017
Joseph, George G. "The crest of Peacock", Princeton University Press, 2000 (Supplement to non-European history of Linear Algebra)
Kolman, Bernard and Hill, David R., "Elementary Linear Algebra", 9th edition, Saunders College Publishing, 2007 (Only for instructors; thorough exposition; the most recent edition)
Strang, Gilbert. "Linear Algebra and its Applications", 4th edition, Saunders College Publishing, 2006.
Gilbert, Strange Introduction to Linear Algebra, 4th edition, Wellesley-Cambridge Press, 2009.
Carlson, David, et al, editors. Resources for Teaching Linear Algebra. Mathematical Association of America, 1997.
Carlson, David; Johnson, Charles R.; Lay, David C.; Porter, Duane A., editors. Linear Algebra Gems: Assets for Undergraduate Mathematics. Mathematical Association of America, 2002.

Learning Outcomes and Objectives

Course Objectives

  • Solve and analyze systems of linear equations using matrices and matrix theory
  • Investigate special matrices and matrix operations including powers and factorization
  • Develop understanding and use of n-dimensional vectors and vector operations
  • Define and investigate vector spaces and vector sub-spaces and find their bases and dimensions
  • Establish understanding of linear transformations and their geometry and find their matrix representation
  • Define eigenvalues and eigenvectors and use them to diagonalize square matrices and solve related problems
  • Utilize methods of linear algebra to solve application problems selected from engineering, science and related fields
  • Prove basic results in linear algebra using appropriate proof-writing techniques
  • Analyze the theory and application of Linear Algebra through projects, extended reading, or programming and computational problems.

CSLOs

  • Construct and evaluate linear systems/models to solve application problems.

  • Solve problems by deciding upon and applying appropriate algorithms/concepts from linear algebra.

  • Apply theoretical principles of linear algebra to define properties of linear transformations, matrices and vector spaces.

Outline

  1. Solve and analyze systems of linear equations using matrices and matrix theory
    1. Convert systems of equations to matrix equations and produce augmented and coefficient matrices.
    2. Use row operations to put matrices into row echelon and row reduced echelon forms
    3. Apply the row echelon form of a matrix to classify a system of linear equations as consistent/inconsistent, dependent/independent.
    4. Use row reduced form of augmented matrices to write solutions in vector and parametric forms.
    5. Examine the condition number of a matrix and determine its affect on the inaccuracy of approximate solutions to linear systems
    6. Investigate and solve problems from geometry, science, engineering as well as problems that explore multi-cultural perspectives and problems from fields of interest to students
  2. Investigate special matrices and matrix operations including powers and factorization
    1. Find sums, scalar multiples of matrices
    2. Find products of matrices using point by point, column and row multiplication methods
    3. Find the transpose of a matrix
    4. Define and compute the inverse of a square matrix
    5. Solve systems of equations using the inverse of the coefficient matrix and establish conditions for its invertibility
    6. Define and investigate basic properties of triangular, diagonal and symmetric matrices
    7. Define the determinant of a square matrix and study the properties of determinants including triangular, diagonal and invertible matrices
    8. Find determinants of square matrices using cofactor expansion, row and column operations
    9. Define and use elementary matrices and use them to factor square matrices into a product of lower and upper triangular matrices and to find the inverse of a matrix
    10. Use determinants to solve and analyze square systems of equations
    11. Solve systems of linear equations using LU factorization and forward and backward substitution. (Optional)
  3. Develop understanding and use of n-dimensional vectors and vector operations
    1. Explore n-dimensional vectors and basic vector operations
      1. Find the magnitude of a vector
      2. Define and compute direction vectors
      3. Find sums and differences and scalar multiples of vectors
      4. Define and find inner and cross product of vectors
      5. Use vector inner product to determine angles between two vectors and orthogonality
    2. Apply the algebra of 2D and 3D vectors to study lines and planes in 3D space.
      1. Find the equation of a plane
      2. Find the equation of a line
      3. Define vector projection and find the projection of one vector onto another
      4. Find the distance between a point and a plane
      5. Find the distance between a point and a line
  4. Define and investigate vector spaces and vector sub-spaces and find their bases and dimensions
    1. Develop an understanding of Euclidean n-dimensional space, norm, Cauchy-Schwartz and triangle inequalities
    2. Investigate general linear spaces and subspaces such as but not limited to the space of continuous functions
    3. Define linear dependence and independence of vectors in general vector space setting and determine linearity by
      1. use of the definition
      2. use of the Wronskian
    4. Find bases and dimensions of vector spaces.
    5. Express vectors as a linear combinations of a set of basis vectors
    6. Change basis and investigate change of bases matrices.
    7. Use the Gram-Schmidt process to produce an orthonormal set of vectors.
    8. Solve problems using basis and orthonormal basis of general vector spaces
    9. Apply the Gram-Schmidt process to investigate special polynomials (like Legendre) (optional)
  5. Establish understanding of linear transformations and their geometry and find their matrix representation
    1. Define linear transformations on general vector spaces and find their domains and ranges
    2. Interpret linear transformations in 2-and 3-space as geometric operations such as but not limited to translations, rotations, dilation, reflections, and projections on vector subspaces
    3. Study one to one and onto linear transformations
    4. Construct matrices of general linear transformations using non-standard bases.
    5. Define the four fundamental subspaces of linear transformations
    6. Investigate and find nullity and rank of linear transformations
    7. Construct bases of the four fundamental subspaces of a matrix and use them to solve problems in 2- and 3-space
    8. Find composition and inverse of linear transformations and use them to find images of vectors in 2- and 3-space
  6. Define eigenvalues and eigenvectors and use them to diagonalize square matrices and solve related problems
    1. Define eigenvalues and eigenvectors of a matrix and explore their geometric interpretation.
    2. Use the characteristic equation to find the eigenvalues of a matrix
    3. Find the eigenvectors of a matrix
    4. Determine the geometric and algebraic multiplicities of eigenvalues
    5. Find the eigenspace of a matrix
    6. Investigate conditions for both diagonalization and orthogonal diagonalization of a matrix
    7. Use standard procedures to both diagonalize and orthogonally diagonalize matrices
    8. Choose application problems from areas such as dynamical systems, Markov chains, cryptography, and game theory as well as problems that explore multi-cultural perspectives and problems from fields of interest to students
  7. Utilize methods of linear algebra to solve application problems selected from engineering, science and related fields
    1. Iterative methods for solving linear systems such as Gauss-Seidel method.
    2. The power method for finding eigenvalues of a matrix and its application to internet search engines.
    3. Use of projection matrices for the general least squares approximations.
    4. Transform equations of general quadric surfaces into standard forms
  8. Prove basic results in linear algebra using appropriate proof-writing techniques
    1. Linear dependence and independence
    2. Linearity
    3. Properties of subspaces
    4. Properties of eigenvalues and eigenvectors
    5. injectivity (One to one) and surjectivity (onto) of functions and linear operators
    6. Other proofs of statements, as deemed necessary, to improve students understanding of course content.
  9. Analyze the theory and application of Linear Algebra through projects, extended reading, or programming and computational problems.
    1. Typical problem solving topics may include any of the following:
      1. Numeric analysis of the efficiency and error for algorithms covered
      2. Volume of solids of revolution about lines that are not horizontal or vertical
    2. Typical applied projects may include any of the following:
      1. Derivation of some of the formulas in statistics and probability
      2. Applications of integral calculus in other disciplines such as biology chemistry, and economics.
      3. Details and history of the proofs for some of the main theorems in integral calculus
Back to Top