Chabot College
Fall 2004
Replaced Fall 2010
Course Outline for Mathematics 6
ELEMENTARY LINEAR ALGEBRA
Catalog Description:
6 - Elementary Linear Algebra
3 units
Introduction to linear algebra: matrices, determinants, systems of equations, vector spaces, linear
transformations, eigenvalue, eigenvectors, applications. Prerequisites: Mathematics 2 (completed with a
grade of C or higher). 3 hours.
[Typical contact hours: 52.5]
Prerequisite Skills:
Upon entering the course, the student should be able to:
1.
define the natural logarithmic function in terms of a Riemann integral;
2.
integrate and differentiate logarithmic functions;
3.
define and differentiate inverse functions;
4.
define an exponential function;
5.
differentiate and integrate exponential functions;
6.
differentiate and integrate inverse trigonometric functions;
7.
differentiate and integrate hyperbolic functions and their inverses;
8.
solve application problems involving logarithmic, exponential, inverse trigonometric, and hyperbolic
functions;
9.
solve differential equations using separation of variables;
10.
use standard techniques of integration such as integration by parts, trigonometric integrals,
trigonometric substitution, partial fractions, rational functions of sine and cosine;
11.
graph polar equations and find area of regions enclosed by the graphs of polar equations;
12.
evaluate limits using L'Hopital's Rule;
13.
evaluate improper integrals;
14.
use parametric representations of plane curves;
15.
perform basic vector algebra in R2 and R3 and interpret the results geometrically;
16.
find equations of lines and planes in R3;
17.
construct polynomial approximations (Taylor polynomials) for various functions and estimate their
accuracy using an appropriate form of the remainder term in Taylor's formula;
18.
determine convergence of sequences;
19.
determine whether a series converges absolutely, converges conditionally or diverges;
20.
construct (directly or indirectly) power series representations (Taylor series) for various functions,
determine their radii of convergence, and use them to approximate function values.
Expected Outcomes for Students:
Upon completion of the course, the student should be able to:
1.
2.
3.
4.
5.
6.
solve systems of linear equations using Gaussian elimination, matrix, and determinant techniques;
compute determinants of all orders;
perform all algebraic operations on matrices and be able to construct their inverses, adjoints,
transposes; determine the rank of a matrix and relate this to systems of equations;
recognize and use the properties of vector spaces and inner product spaces, and understand the
concepts of subspaces, linear independence, bases, orthogonality, and their relation;
identify and use the properties of linear transformations and their relation to matrices;
solve eigenvalue problems, diagonalize and orthogonally diagonalize matrices.
Course Content:
1.
2.
Solution of systems of linear equations by various techniques
The algebra of matrices; inverses, transposes, determinants
Chabot College
Course Outline for Math 6, page 2
Fall 2004
3.
Vector spaces, subspaces, linear independence, bases, dimension, row and column space, inner
product spaces, orthonormal bases, Gram-Schmidt Process
Linear transformations, their properties and matrix representations, geometry of linear
transformation, change of basis
Eigenvalue problems, similarity, orthogonal bases, and diagonalizing matrices
Applications: quadratic forms, the Principal Axes Theorem, approximation, Fourier series (if time
permits)
4.
5.
6.
Methods of Presentation:
1.
2.
Lectures/demonstrations
Discussion
Assignments and Methods of Evaluating Student Progress:
1.
Typical Assignments
a. Find the standard matrix for the stated composition of linear operators on R2.
1) A rotation of 90, followed by a reflection about the line y = x.
2) An orthogonal projection on the y-axis, followed by a contraction with factor k = 0.5.
3) A reflection about the x-axis, followed by a dilation with factor k = 3.
b. Let V be an inner product space. Show that if w is orthogonal to both u1 and u2, it is orthogonal
to k1u1 + k2u2 for all scalars k1 and k2. Interpret this result geometrically in the case where
V is R3 with the Euclidean inner product.
2.
Methods of Evaluating Student Progress
a. Quizzes
b. Exams and final examination
c. Homework
Textbook(s) (Typical):
Elementary Linear Algebra, Howard Anton, John Wiley Publishers, 2000
Special Student Materials:
A calculator may be required.
CS:al
Revised: 10/2003