Elementary Linear Algebra
 Course: MAT225 First Term: 2021 Fall Final Term: Current Final Term: 9999 Lecture   3.0 Credit(s)   3.0 Period(s)   3.0 Load       Credit(s)    Period(s)    Load Subject Type: Academic Load Formula: S - Standard Load

Description: Introduction to matrices, systems of linear equations, determinants, vector spaces, linear transformations and eigenvalues. Emphasizes the development of computational skills.

MCCCD Official Course Competencies
1. Apply matrices to solve a system of linear equations. (I)
2. Analyze the existence and nature of the solution of a system of linear equations using the determinant of an appropriate matrix. (I, II)
3. Use current technology to solve problems within the context of the course. (I, II, III, IV, V, VI)
4. Write the solution of a system of linear equations as a linear combination of vectors. (I, III)
5. Determine if a set of vectors forms a vector space and find a basis. (III)
6. Determine the dependence of a set of vectors. (III)
7. Identify the four fundamental subspaces of a matrix. (III)
8. Construct an orthonormal set of vectors by using the Gram-Schmidt process. (IV)
9. Find eigenvalues and eigenvectors of a square matrix. (V)
10. Define a linear transformation and its range. (VI)
11. Find the Kernel of a linear transformation. (VI)
12. Analyze linear algebra real world applications. (VII)
MCCCD Official Course Competencies must be coordinated with the content outline so that each major point in the outline serves one or more competencies. MCCCD faculty retains authority in determining the pedagogical approach, methodology, content sequencing, and assessment metrics for student work. Please see individual course syllabi for additional information, including specific course requirements.

MCCCD Official Course Outline
I. Systems of linear equations and matrices
A. Linear systems
1. Methods of elimination and substitution
2. Classification of systems of equations and their solutions
B. Matrices
1. Operations on matrices
2. Properties of matrix operations
3. Inverse of a matrix
C. Matrix representation of linear systems
1. Solutions of equations using matrices
2. Reduced row echelon form
3. LU factorization
II. Determinants
A. Definitions and properties
B. Cofactor expansion
C. Cramer`s rule
III. Vectors and vector spaces
A. Vectors in R2 and R3
1. Vector operations
2. Orthogonal and unit vectors
B. Vectors in Rn
1. Inner product
2. Triangle inequality
C. General vector spaces and subspaces
1. Properties of vector spaces
2. Definition of a subspace
3. Span of a set of vectors
D. Linear independence
1. Linear combination of vectors
2. Span of a set of vectors
E. Basis and dimension
1. Definition of a basis
2. Finite and infinite - dimensional vector spaces
3. Change of basis
F. The four fundamental subspaces
1. Rank of a matrix
2. Null space definition and properties
IV. Orthogonality
A. Gram-Schmidt process
B. Orthonormal basis
C. Least squares
V. Eigenvalues and Eigenvectors
A. Characteristic polynomials and equations for square matrices
B. Determining eigenvalues and eigenvectors for a square matrix
C. Matrix factorization
1. QR Factorization
2. Matrix diagonalization
VI. Linear Transformations and Matrices
A. Properties and examples of linear transformations
B. Kernel and range of a linear transformation
C. Matrix representation of a linear transformation
VII. Application of linear algebra techniques to real world scenarios
A. Principal component analysis
B. Singular value decomposition
C. Leslies matrices
D. Markov chains
E. Additional topics per instructor discretion

MCCCD Governing Board Approval Date: February 23, 2021

All information published is subject to change without notice. Every effort has been made to ensure the accuracy of information presented, but based on the dynamic nature of the curricular process, course and program information is subject to change in order to reflect the most current information available.