# Mathematics 3301-010 Linear Algebra

## Course Description

Designed for students planning studies in mathematics, engineering, computer
science, and physics. Topics

for the course include: systems of linear equations, vectors, matrices, linear
transformations, vector spaces,

eigenvalues/eigenvectors/eigenspaces/eigenbasis, orthogonalization and
diagonalization.

**1) Systems of linear equations**

a) Represent a linear system as an augmented matrix and use Gaussian
elimination find the solution (s) to

the system. Develop the notions of pivot columns, free variables, row echelon
form, and reduce row echelon

form of a matrix.

b) For linear systems two and three unknowns develop an understand the geometry
of a linear system and

its solution(s).

c) Use a computer algebra system (CAS) to solve linear system of any size using
Gaussian elimination/row

reduction.

**2) Vectors**

a) Develop a visual understanding of vectors in R_{2} and R_{3}.

b) Develop the notion of a linear combination of vectors algebraically and
geometrically along with the span

of a set of vectors.

c) Develop the notions of the vector form of a line, the vector form of a plane,
and the decomposition of the

solution to a linear system as a linear combination of vectors.

d) Develop thoroughly the notion of linear independence of a set of vectors.
Emphasize the meaning of linear

independence geometrically for vectors in R_{2} and R_{3}.

**3) Matrices**

a) Develop the notion of a matrix times a vector as being
a linear combination of the columns of the matrix

using the vector entries as weights.

b) Establish the connections between linear systems, vector equations, and
matrix equations. Specifically the

equivalent representation of a linear system as a vector equation or a matrix
equation.

c) Develop matrix multiplication by generalizing the matrix-vector product to
the matrix-matrix product.

d) Develop fully the rules of matrix algebra.

e) Develop fully the notion of multiplicative inverses for matrices. When does
matrix have an inverse and

how is the inverse of a matrix determined?

f) Develop competence in using a CAS to work with matrices, vectors, etc.

g) Introduce the “Invertibility Theorem” establishing equivalences amongst
matrices, vectors, and solutions to

linear systems.

**4) Linear Transformations**

a) Define a linear transformation from R_{m} to R_{n}.
Establish that every linear transformation is represented

by a matrix transformation. Develop the linear properties and algebraic
operations on linear transformations.

Define the kernel and range of a linear transformation.

b) Investigate fully the geometric linear transformations in R_{2} and R_{3}:
dialations/contractions, shears,

rotations, reflections, rotations, and projections.

c) Develop competence in using a CAS to work with linear transformations.

**5) Vector Spaces**

a) Develop the notions of vector spaces, subspaces, basis,
dimension for R_{n} vector spaces. Emphasize

geometry for R_{2} and R_{3}. Develop the notion of coordinates
relative to a basis and how to change between

different basis for a vector space.

b) Develop the subspaces associated with a matrix – row space, column space, and
null space. Include

techniques for finding a basis for each of these spaces.

**6) Determinants**

Development of the basic properties of determinants and how to find the determinant of a matrix using a CAS.

**7) Eigenvalues, eigenvectors, eigenspaces, and
eigenbasis**

a) Develop the notions of the eigenvalues, eigenvectors,
and eigenspaces of a matrix and emphasize the

geometry for 2 by 2 and 3 by 3 matrices.

b) Define and develop the properties of the characteristic polynomial of a
matrix .

c) Develop the notion of diagonalization of a matrix and similarity.

d) Develop competence in using a CAS to find eigenvalues and eigenvectors for a
matrix, to determine the

characteristic polynomial of a matrix, and to diagonalize a matrix.

**8) Orthogonality**

a) Develop the notion of orthogonality for a set of
vectors and an orthogonal/orthonormal basis for R_{n}.

b) Develop the notion of orthogonal projections (into a subspace) and the
Gram-Schmidt process for finding

an orthogonal basis of a subspace.

c) Develop the notion of orthogonal matrices, orthogonal complements, and
orthogonal projections.

d) Use orthogonality to develop least -squares solutions to inconsistent linear
systems. Include the normal

equations to the linear system. Use least-squares solutions to solve a data
fitting problem.

e) Develop competence in using a CAS to work with orthogonality concepts,
Gram-Schmidt, and finding

least-squares solutions to inconsistent linear systems.

Advanced math courses at WWCC:

## Required Materials

1. The textbook for the course is **Visual Linear Algebra**,
1st edition, Herman/Pepe/Schulz, John Wiley &

Sons.

2. Access to the computer algebra system Mathematica. WWCC’s site license
provides students their own

personal copies of Mathematica 7. Install DVD’s are available in the WWCC
Bookstore for $1.99.

3. Engineering Computation paper for homework assignments. Available in the WWCC
Bookstore for

approximately $2.50 for a 100-sheet pad.

## Attendance

Attendance at every class session is expected. I
understand absences are sometimes unavoidable and will

work with students when such occasions arise. In the event of an absence
occurring on the date of a

scheduled exam or quiz, **prior** arrangements must be made in order to
schedule another time to write the

exam.

## Cell Phones/PDAs

Our classroom is a **No Cell Phone/PDA environment.**
Cell phones are to be silenced before class begins

and put away. Cell phones/PDAs are not to be accessed for any reason during
classtime. Text messaging is

not allowed during class. Using a cell phone/PDA as a calculator is not
acceptable - you should have a

scientific calculator for use in the course. Develop the habit of silencing your
phone when entering the

classroom - I'm confident that everyone can manage to go 50 minutes without
accessing their cell phone!

Our classroom is equipped with computers. We will use the
computer resources at various times throughout

the course for course activities. Other use of the computers is not to occur
during class - no checking email,

no instant message, no web browsing, no gaming, and no working on online
homework assignments during

class time.

## Homework Assignments

Homework assignments comprised of textbook problems will given regularly and discussed in class.

## Exams

There will be 4 exams: 3 exams scheduled during the
quarter and a comprehensive final exam. Each exam

will be composed of an in-class portion and a take-home portion that is to be
completed outside of class and

will require the use of Mathematica. The final exam is worth 200 points and the
other 3 exams are worth

150 points each.

## Grades

Grades for the course are computed by adding up the number
of points earned and dividing by the total

number of points possible in the course. Final grades are simply a function of
the percentage of possible

points earned. Let p be the percent of the possible course points earned by a
student, the course grade is

then given in the following table:

93% <= p<=100% ->A

90% <= p <93% -> A-

87% <= p <90% -> B+

83% <= p < 87% -> B

80% <= p < 83% -> B-

77% <= p < 80% -> C+

73% <= p < 77% -> C

70% <= p < 73% -> C-

67% <= p < 70% -> D+

60% <= p < 67% -> D

0% <= p < 60% -> F

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