Yuba Community College District

Yuba College Course Outline

### Course Information

Course Number: MATH 3
Full Course Title: Linear Algebra
Short Title: Linear Algebra
TOP Code: 1701.00 - Mathematics, General
Effective Term: Fall 2013

### Course Standards

Course Type: Credit - Degree Applicable
Units: 3.0
Total class hours: 162.0
Total contact hours in class: 54.0
Lecture hours: 54.0
Hours outside of class: 108.0
Repeatable: No

### Minimum Qualifications for Instructors

• Mathematics (Masters Required)

### Course Description

This course develops the techniques and theory needed to solve and classify systems of linear equations. Solution techniques include row operations, Gaussian elimination, and matrix algebra. Investigates the properties of vectors in two and three dimensions, leading to the notion of an abstract vector space. Vector space and matrix theory are presented including topics such as inner products, norms, orthogonality, eigenvalues, eigenspaces, and linear transformations. Selected applications of linear algebra are included.

### Conditions of Enrollment

Satisfactory completion of: MATH 1B Recommended successful completion of Math 1C. Math 1C introduces vectors and a large part of Math 3 will be about vectors and vector spaces. Moreover, Math 1C (formerly Math 2A) was the prerequisite for Math 3 before the revision.

• Language - recommended eligibility for English 1A
• Mathematics - recommended eligibility for Math 52

### Content

Course Lecture Content
1. Techniques for solving systems of linear equations including Gaussian and Gauss-Jordan elimination and inverse matrices
2. Matrix algebra, invertibility, and the transpose
3. Relationship between coefficient matrix invertibility and solutions to a system of linear equations and the inverse matrices
4. Special matrices: diagonal, triangular, and symmetric
5. Determinants and their properties
6. Vector algebra for Rn
7. Real vector space and subspaces
8. Linear independence and dependence
9. Basis and dimension of a vector space
10. Matrix-generated spaces: row space, column space, null space, rank, nullity
11. Change of basis
12. Linear transformations, kernel and range, and inverse linear transformations
13. Matrices of general linear transformations
14. Eigenvalues, eigenvectors, eigenspace
15. Diagonalization including orthogonal diagonalization of symmetric matrices
16. Inner products on a real vector space
17. Dot product, norm of a vector, angle between vectors, orthogonality of two vectors in Rn
18. Angle and orthogonality in inner product spaces
19. Orthogonal and orthonormal bases: Gram-Schmidt process
20. Method of least squares

### Objectives

1. Find solutions of systems of equations using various methods appropriate to lower division linear algebra.

2. Use bases and orthonormal bases to solve problems in linear algebra.

3. Find the dimension of spaces such as those associated with matrices and linear transformations.

4. Find eigenvalues and eigenvectors and use them in applications. **Requires Critical Thinking**

5. Prove basic results in linear algebra using appropriate proof-writing techniques such as linear independence of vectors; properties of subspaces; linearity, injectivity and surjectivity of functions; and properties of eigenvectors and eigenvalues. **Requires Critical Thinking**

### Student Learning Outcomes

1. Computation – Solve a system of linear equations using matrix methods.
2. Computation – Calculate eigenvalues and eigenvectors for a 3x3 matrix.
3. Critical Thinking – Formulate transformations between n-dimensional vector spaces.

### Methods of Instruction

• Lecture/Discussion

### Assignments

Writing Assignments

### Methods of Evaluation

• Exams
• Homework
• Portfolio
• Problem Solving Exercises
• Quizzes
• Other
Group or individual projects.

### Course Materials

Textbooks:
1. Gilbert Strang. Introduction to Linear Algebra, 4th ed. Wellesley Cambridge Press, 2009, ISBN: 978-0-980232-71-4
Equivalent text is acceptable