SANTA BARBARA CITY COLLEGE
ASSOCIATE
DEGREE CREDIT COURSE OUTLINE
Department: Mathematics
Subject Area and Course Number: Mathematics 210
Course Title: Linear Algebra
Discipline: Mathematics
Units: 4
Repeatability: None
Catalog Course Description: Finite dimensional vector spaces,
linear independence, bases, systems of linear equations, linear
transformations, matrices, LU factorization, change of bases, similarity of
matrices, eigenvalues and eigenvectors, diagonalization, applications,
quadratic forms, symmetric and orthogonal matrices, canonical forms,
introduction to infinite dimensional vector spaces.
Description for Schedule of Classes:
Finite dimensional vector spaces, linear independence, bases, systems of linear
equations, linear transformations, matrices, LU factorization, change of bases,
similarity of matrices, eigenvalues and eigenvectors, diagonalization,
applications.
Lecture Hours per Week: 4.3 (64-72
Total Semester Hours)
Laboratory Hours per Week: None
Plus Hours: None
Prerequisites: Math 160, with a grade of "C"
or better.
Co-requisites: None
Skills Advisories: None
Course Advisories: None
Limitation on Enrollment: None
Course Objectives: Related to the use of matrices and the
underlying structure of vector spaces.
1.
Solve Ax=b by elimination, the
invertibility of A, or LU factorization.
2.
Determine basis and dimension of
vector spaces, especially for column and null spaces of A and AT.
3.
Project a vector onto a subspace.
4.
Use the Gram-Schmidt procedure to
factor a matrix into QR form.
5.
Compute determinants using their
properties.
6.
Find eigenvalues and eigenvectors of
a matrix.
7.
Diagonalize a matrix, compute its
powers, and its exponential matrix.
8.
Write quadratic forms as matrix
products xTAx using real, symmetric matrices A.
9.
Apply Linear Algebra to problems
such as systems of differential equations, finite difference equations, graphs
and networks, Markov matrices, Fourier matrix, Fast Fourier Transform, and
linear programming.
Course Content and Scope
1. Systems
of Linear Equations and Matrices
a. Gaussian
Elimination and LU factorization
b. Homogeneous
Systems of Linear Equations
c. Matrices
and Matrix Operations
d. Rules
of Matrix Arithmetic
e. Elementary
Matrices and a Method for finding A-1
f. Further
Results on Systems of Equations and Invertibility
2. Determinants
a. The
Determinants by Row Reduction
b
. Evaluating
Determinants by Row Reduction
c. Properties
of the Determinant Function
d. Cofactor
Expansion; Cramer's Rule
3. Vectors
in 2-Space and 3-Space
a. Introduction
to Vectors (Geometric)
b. Norm
of a Vector; Vector Arithmetic
c. Dot
Product; Projections
d. Cross
Product
e. Lines
and Planes in 3-Space
4. Vector
Spaces
a. Euclidean
n-Space
b. General
Vector Spaces
c. Subspaces
d. Linear
Independence
e. Basis
and Dimension
f. Row
and Column Space; Rank; Finding Bases
g. Inner
Product Spaces
h. Length
and Angle in Inner Product Spaces
i. Orthonormal
Bases; Gram-Schmidt Process; QR factorization
j. Coordinates;
Change of Basis
5. Linear
Transformations
a. Definition
of Linear Transformation
B. Properties
of Linear Transformations; Kernel and Range
c. Linear
Transformations from Rn to Rm
d. Matrices
of Linear Transformations
e. Similarity
6. Eigenvalues,
Eigenvectors
a. Definitions
and Introduction
b. Diagonalization
c. Orthogonal
Diagonalization; Symmetric Matrices
7. Applications
a. Approximation
Problems; Fourier Series
b. Quadratic
Forms
c. Other
applications as time and interest allow
Methods of Instruction: Lecture is the primary activity, along
with student problem solving. Students are expected to work outside of class on
reading the text and on assigned exercises that may include the use of a
computer algebra system, and supplemental reading as determined by the
instructor.
Required Assignments
1. Appropriate
Readings: Students are required to
read assigned chapters in texts.
2. Writing
Assignments: Students must work
assigned mathematical problems requiring the understanding of abstract ideas.
3. Appropriate
Outside Assignments: Students will
be expected to spend a sufficient amount of time outside of class to practice
techniques taught during class time, read assigned materials, and complete
frequent homework and computer assignments.
4. Appropriate
Assignments that Demonstrate Critical Thinking: Students must demonstrate mathematical skills which involve
analyzing information, recognizing concepts in new contexts, and drawing
analogies. They must also analyze logical arguments for validity and write
proofs of their own using both inductive and deductive reasoning within a
logical system. Students will also learn to use software tools in solving
problems from linear algebra.
Methods of Evaluation: A student's grade will be based on
multiple measures of performance in the solving of problems, designing of
mathematical models, preparation and analysis of graphs, and analysis of
logical arguments. Such measures
will include at least four one-hour exams and a comprehensive final examination
requiring demonstrations of problem solving skills. In addition, instructors may make use of quizzes, written
homework assignments, or other appropriate means to judge a student's dexterity
with mathematical skills and familiarity with mathematical vocabulary and
methods of proof.
Appropriate Texts and Supplies
Herman, Pepe, Visual Linear Algebra with Maple and Mathematical Tutorials, Wiley, 2005
Strang, Linear Algebra and its Applications, 4th Ed., Cengage, 2006
Student Learning Outcomes:
1.
Solve
Ax=b using a variety of methods such as Gaussian elimination and inverting the
matrix A.
2.
Identify
a linear transformation and find the eigenvalues and corresponding eigenspaces.
3.
Diagonalize
a matrix or determine why it is not possible to do so.
4.
Orthogonalize
bases using the Gram-Schmidt process and produce unique representations in
terms of these bases.
5.
Prove
Linear Algebra theorems and corollaries.
6.
Apply
Linear Algebra to problems in the sciences.
JK/mej
Approved
December 4, 2006
Revised
5/13/09; 8/24/09
FRC
(WPC)