Discipline: Mathematics
Originator: James Namekata
Riverside Community College District
Integrated Course Outline of Record
Mathematics
3
MAT-3 : Linear Algebra |
College:
Lecture Hours:
54.000 Outside-of-Class Hours: 108.000 Units: 3.00 Grading Methods: Pass/No Pass Letter Grade |
Course Description
Prerequisite:
MAT-1B
Course Credit Recommendation:
Degree Credit
This course examines elementary vector space concepts and geometric interpretations and develops the techniques and theory to solve and classify systems of linear equations. Solution techniques include Gaussian and Gauss-Jordan elimination, Cramer's rule and inverse matrices. Investigates the properties of vectors in two, three and finite dimensions, leading to the notion of an abstract vector space. Vector space and matrix theory are presented including topics such as determinants, linear independence, bases and dimension of a vector space, linear transformation and their matrix representations, inner products, norms, orthogonality, eigenvalues, eigenvectors, and eigenspaces. Selected applications of linear algebra are included. 54 hours lecture. (Letter Grade, or Pass/No Pass option.)
Short Description for Class Schedule
Introduction to matrix algebra with vector spaces and linear transformations.
Entrance Skills:
Before entering the course, students should be able to demonstrate the following skills:- Evaluate definite and indefinite integrals.
Student Learning Outcomes:
Upon successful completion of the course, students should be able to demonstrate the following skills:- Solve systems of linear algebraic equations using various methods including Gaussian and Gauss-Jordan elimination, Cramer's rule and inverse matrices.
- Calculate and apply determinants to a variety of problems including but not limited to areas, volumes, and cross products.
- Determine the rank and the dimension of the kernel for a matrix operator.
- Find bases for vector spaces including but not limited to spaces associated with matrices and linear transformations. Use bases and orthonormal bases to solve problems in linear algebra.
- Find Eigenvalues and related Eigenvectors for a square matrix and use them in applications.
- Use the Gram-Schmidt process to produce an orthonormal basis and use it in applications such as diagonalization of a square matrix.
- Prove basic results in linear algebra using appropriate proof writing techniques.
Course Content:
- Systems of Linear Equations
- Gaussian and Gauss-Jordan Elimination, Cramer's rule, Applications
- Relationship between coefficient matrix invertibility and solutions to a system of linear equations and the inverse matrices
- Matrices
- Operations, Transpose, Properties, Inverses, Applications
- Special Matrices: Diagonal, triangular, and symmetric
- Determinants
- Evaluation, Properties, Applications
- Vector Spaces
- Real Vector Space and Subspaces, Linear Independence, Basis and Dimension, Applications
- Matrix-generated spaces: row space, column space, null space, rank, nullity
- Vector algebra for R^{n}
- Change of basis
- Inner Product Spaces
- Dot products, norm of a vector, angle between vectors, orthogonality of two vectors in R^{n}
- Inner products on a real vector spaces, angle and orthogonality in inner product spaces
- Orthogonal and orthonormal bases: Gram Schmidt Process
- Linear Transformations
- Kernal and Range, Matrices for Linear Transformations and Inverse Linear Transformations, Similarity, Applications
- Eigenvalues, Eigenvectors, and Eigenspaces
- Diagonalization, Applications
- Proofs of Fundamental Theorems in linear algebra
Methods of Instruction:
Methods of instruction used to achieve student learning outcomes may include, but are not limited to, the following activities:
- Class lectures, discussions, and demonstrations of appropriate proof writing techniques, various methods in solving systems of linear algebraic equations, methods in calculating determinants, rank and the dimension of the kernel for a matrix operator, methods in finding bases for vector spaces, Eigenvalues and related Eigenvectors for a square matrix.
- Drills and pattern practices utilizing hand-outs and/or computer-based tools in order to assist the students in mastering the techniques involved in applying determinants to find areas, volumes and cross products and techniques to produce orthonormal bases and find solutions of systems of linear algebraic equations.
- Provision and employment of a variety of learning resources such as videos, slides, audio tapes, computer-based tools, manipulatives, and worksheets in order to address multiple learning styles and to reinforce material.
- Pair and small group activities, discussions, and exercises in order to promote mathematics discovery and enhance problem solving skills.
Methods of Evaluation:
Students will be evaluated for progress in and/or mastery of student learning outcomes using methods of evaluation which may include, but are not limited to, the following activities:
- Evaluation of written homework assignments and/or computerized homework assignments for correct application of various methods in solving systems of linear algebraic equations, methods in calculating determinants, rank and the dimension of the kernel for a matrix operator, methods in finding bases for vector spaces, Eigenvalues and related Eigenvectors for a square matrix as well as the correct use of symbols and vocabulary of linear algebra.
- Evaluation of quizzes and midterm/final examinations for conceptual understanding as well as correct technique and application of various methods in solving systems of linear algebraic equations, methods in calculating determinants, rank and the dimension of the kernel for a matrix operator, methods in finding bases for vector spaces, Eigenvalues and related Eigenvectors for a square matrix as well as use appropriate proof writing techniques in proofing basic results in linear algebra.
- Assessment of classroom discovery activities for content knowledge and conceptual understanding.
Sample Assignments:
Outside-of-Class Reading Assignments
- Read and analyze text, examples, and notes covering topics such as employ gaussian elimination to find bases of a vector space and apply determinants to a variety of problems including finding areas, volumes, cross products and solving systems of linear equations.
Outside-of-Class Writing Assignments
- Write proofs of statements in linear algebra using definitions and previously proven lemmas, corollaries, and theorems.
Other Outside-of-Class Assignments
- Problem sets that require students to perform Gaussian elimination on systems of linear equations and use the Gram-Schmidt process to produce an orthonormal basis.
Course Materials:
All materials used in this course will be periodically reviewed to ensure that they are appropriate for college level instruction. Possible texts include the following:
- Anton, Howard. Elementary Linear Algebra. 10th Wiley, 2010.
- Larson. Elementary Linear Algebra. 7th Brooks/Cole, 2013.
- Strang, Gilbert. Introduction to Linear Algebra. 4th Wellesley-Cambridge Press, 2009.
Codes/Dates:
CB03 TOP Code:
1701.00 - Mathematics, General
CB05 MOV Transfer Status: Transfers to Both UC/CSU (A)
CB05 NOR Transfer Status: Transfers to Both UC/CSU (A)
CB05 RIV Transfer Status: Transfers to Both UC/CSU (A)
C-ID#: MATH 250
CB05 MOV Transfer Status: Transfers to Both UC/CSU (A)
CB05 NOR Transfer Status: Transfers to Both UC/CSU (A)
CB05 RIV Transfer Status: Transfers to Both UC/CSU (A)
C-ID#: MATH 250
Board of Trustees Approval Date:
04/16/2013
COR Rev Date:
04/16/2013