### MATH 2641

This is an archive of the Common Course Outlines prior to fall 2011. The current Common Course Outlines can be found at http://www.gpc.edu/programs/Common-Course-Outlines.**Credit Hours**3

**Course Title**Linear Algebra

**Prerequisite(s)**MATH 2432 (may also be co-requisite)

**Corequisite(s)**

*None Specified*

**Catalog Description**

This course is designed to introduce the student to the basic notions of linear algebra.

Topics include matrices, systems of linear equations, vector spaces, inner products,

bases, linear transformations, eigenvalues, and eigenvectors.

**Expected Educational Results**

As a result of completing this course, the student will be able to do the following:

1. Use Gauss-Jordan Elimination to solve systems of linear equations and

identify those with one solution, no solution, or an infinite number of solutions.

2. Perform operations with matrices, namely,

a) Matrix addition and subtraction,

b) Scalar multiplication,

c) Transposition,

d) Matrix multiplication,

e) Matrix inversion, and

f) Calculation of rank and determinant.

3. Solve problems that require that the student demonstrate comprehension of

fundamental properties of the above operations including, but not limited to,

a) Axioms for matrix addition and scalar multiplication,

b) Laws for transposes of matrix sums, products, and scalar multiples,

c) Laws for inversion of matrix transposes, products, and scalar multiples,

d) Laws for determinants of matrix products, inverses, and transposes,

e) Significance of the determinant of a matrix yielding zero, and

f) Significance of the product of two matrices yielding the identity.

4. Demonstrate comprehension of fundamental definitions used in the study of

linear algebra by determining whether or not

a) A subset is a subspace of a given vector space,

b) A vector lies in the span of a given set of vectors and, if so, express

that vector as a linear combination of the others,

c) A set of vectors is linearly independent,

d) A set of vectors forms a basis of a given vector space or subspace,

e) A vector function is a linear transformation, and

f) A linear transformation is on-to-one, onto, or an isomorphism.

5. Actively use the above definitions to obtain

a) A basis for and calculation of the dimension of a given vector

space or subspace,

b) A linear transformation that satisfies given properties and a computation

of the action of that transformation on a given vector,

c) A basis for the kernel, a basis for the image, the nullity, and rank of

a given linear transformation,

d) The action of the composition of two given linear transformations (in either order)

on a given vector, and

e) The inverse transformation of a given isomorphism and its action on a given vector.

6. Using an appropriate inner product, obtain

a) The norm of a given vector,

b) The unit vector that is a scalar multiple of a given vector,

c) A basis for and calculation of the dimension of the orthogonal complement of

a given subspace,

d) The projection of a given vector onto a given subspace,

e) An orthogonal basis for a given inner product space or subspace, and

f) An orthonormal basis by normalizing an orthogonal basis.

7. Obtain the characteristic polynomial, eigenvalues, eigenvectors, diagonalization,

and a basis for each eigenspace for a given square matrix.

8. Use least-squares methods to approximate solutions for inconsistent systems.

9. Apply the basic definitions for quadratic forms.

**General Education Outcomes**

I. This course addresses the general education outcome relating to communication by providing

additional support as follows:

A. Students develop their listening and speaking skills through participation in

class and through group problem solving.

B. Students develop their reading comprehension skills by reading the text and by

reading the instructions for text exercises, problems on tests, or on projects.

Reading mathematics text requires recognizing symbolic notation as well as

analyzing problems written in prose.

C. Students develop their writing skills through the use of problems that require

written explanations of concepts.

II. This course addresses the general education outcome of demonstrating effective individual

and group problem-solving and critical-thinking skills as follows:

A. Students must apply mathematical concepts previously mastered to new problems

and situations.

B. In applications, students must analyze problems and describe problems with either

pictures, diagrams, or graphs, then determine the appropriate strategy for

solving the problem.

III. This course addresses the general education outcome of using mathematical concepts to

interpret, understand, and communicate quantitative data as follows:

A. Students must demonstrate proficiency in problem-solving skills including applications

of linear systems, linear transformations and vector space methods.

B. Students must write linear systems to describe real-world situations and interpret

information about the number of solutions as well as the solution of the

systems themselves.

C. Students must solve systems of linear equations which often arise in modeling

numerical relationships.

**Course Content**

1. Operation and Properties of matrices.

2. Vector Spaces and Subspaces.

3. Linear Transformation.

4. Inner Products and Orthogonality.

5. Eigenvalues, Eigenvectors, and Diagonalization of Matrices.

ENTRY LEVEL COMPETENCIES

Upon entering this course the student should be able to do the following:

1. Analyze problems using critical thinking skills.

2. Construct correct expressions using algebraic symbols and notation from statements;

3. Add, subtract, multiply, divide, factor, differentiate, and integrate polynomial

and rational functions.

4. Recognize polynomial and rational functions of a single variable and describe

their domain and range.

5. Perform composition of function and describe the domain and range of the composite.

6. Recognize one-to-one functions; find their inverse, and describe the domain

and range of the inverse.

7. Use set notation correctly.

**Assessment of Outcome Objectives**

I. COURSE GRADE

Exams, assignments, and a final exam prepared by individual instructors will be used to

determine the course grade.

II. DEPARTMENTAL ASSESSMENT

This course will be assessed every three years. The assessment instrument will consist

of a set of free response questions that will be included as a portion of the final

exam for all students taking the course.

A committee appointed by the Executive Committee of the Mathematics Academic Group will

grade the assessment instrument.

III. USE OF ASSESSMENT FINDINGS

The Math 2641 committee, or a special assessment committee appointed by the Executive

Committee of the Mathematics Academic Group, will analyze the results of the assessment

and determine implications for curriculum changes. The committee will prepare a report

for the Academic Group summarizing its finding.

EFFECTIVE DATE: August 1998 APPROVED DATE: April 1998

Reviewed by Committee: April 2005

Last Revised: Aug. 10, 2011

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