MTH
177 Syllabus

**Division**: Arts and
Sciences Date: February 2014

**Curricula in Which Course is Taught**: Science, Liberal
Arts

**Course Number and Title**: MTH 177, Introductory Linear
Algebra

**Credit Hours**: 2 **Hours/Wk
Lecture**: 2 **Hours/Wk Lab**: **Lec****/Lab Comb**: 2

I.
**Catalog
Description: **Covers matrices, vector spaces,
determinants, solutions of systems of linear equations, and Eigen values.
Designed for mathematical, physical, and engineering science programs.

II. **Relationship of the course to curricula objectives in which
it is taught**: This course allows a student to
use logical and mathematical reasoning in problem solving in theoretical and
applied areas that require critical thinking.

III.
**Required background**: Corequisite:
MTH 175

IV. **Course Content**:

A. Introduction to Linear Equations

B.
Row
Reduction of Linear Systems

C.
Consistent
Systems with Infinitely Many Solutions

D.
Inconsistent
Systems

E.
Reduced
Echelon Form

F.
Special
Systems of Equations

G.
Linear Combination
–

H.
Applications
of Linear Equations

I.
Matrix
Addition

J.
Matrix
Multiplication

K.
The
Identity Matrix, The Zero Matrix, and Matrix Powers

L.
Vectors in
Rn and the equation Ax = b

M.
The
General Matrix Equation AX=

N.
Matrix
Inverses

O.
The Transpose
of a Matrix

P.
Theory of
Linear Systems

Q.
Applications
of Matrices

R.
Definition
of a Determinant

S.
Inverses
and Determinants

T.
Properties
of Determinants

U.
Cramer’s
Rule

V. Vectors

W.
Vectors in
the Plane

X.
Algebraic
Vectors; Vector Addition

Y.
Scalar Multiplication
and Unit Vectors

Z.
Rectangular
Coordinates in Three Dimensions

AA.
Vectors in
Three Dimensions

BB.
Dot
Product

CC.
Cross
Product

DD.
Additional
Vector Concepts

EE.
Eigenvectors
and Eigenvalues

FF.The Eigenvalue Problem

GG.
Complex
Numbers and Complex Arithmetic

HH.
Eigenvalues
of Special Matrices

II.
Complex
Eigenvectors

JJ.
Applications
of Eigenvectors

V.
Upon completion of the course the students will be
able to: A.
Define the
basic terminology and methods of solving linear systems, geometric
representation of systems, review of basic matrix terminology, specification of
the coefficient matrix and augmented matrix of the system, recognition of a
linear equation. B.
Apply
elementary row operations, notation and procedure to perform Gauss-Jordon
elimination on a square system with a unique solution, consistent systems
with infinitely manysolutions, and inconsistent
linear systems. C.
Apply
elementary row operations, notation and procedure to perform Gauss-Jordon
elimination on consistent systems with infinitely many solutions, how to
describe these solutions, terms associated with the solution, and dimension. D.
Recognize and
apply elementary row operations, notation and procedure to perform Gaussian
or Gauss-Jordon elimination on inconsistent linear systems, geometric
interpretation for (3x3) systems. E.
Know the
requirements for a system to be in reduced echelon form and using
Gauss-Jordon elimination to transform a matrix to reduce echelon form. Distinction among RRE, REF, and echelon
form for a system. Solve a linear system using the technique of
back-substitution. Determine the rank of a matrix. F.
Recognize if
a system of linear equations is homogeneous or not. Understand the difference
between trivial and nontrivial solutions.
Describe solutions for an m x n linear system. G.
Distinguish
between degenerate and nondegenerate equations.
Identify pivot positions and relationship to rank of a matrix. Understand the
concept of linear combination, linear dependence and independence. H.
Use systems
of linear equations to solve real life problems involving but not limited to
polynomial curve fitting, networking, and electrical networking. I.
Use basic
matrix notation and nomenclature, performing matrix addition and scalar
multiplication using the algebraic properties of these operations. Find the
trace and transpose of a matrix. Recognition of special matrices. J.
Demonstrate
the rules governing matrix multiplication and visualize the size of the
product. K.
Demonstrate
properties and definitions associated with the identity matrix, zero matrix
and powers of matrices. Recognize the irregular aspects of matrix
multiplication. Know the laws of exponents of matrices. L.
Understand
vector notation; represent a system of linear equations as the simple matrix
equation Ax = b, solve matrix equations, and writing the vector form for the
general solution to a system. Write a solution from vector form to linear
combination form. M.
Use of the column
form to represent a matrix and solving the matrix equation AX=B. Solve
several systems having the same coefficient matrix. N.
Define
inverse and when the inverse matrix exists. Use of the determinant test for a
2 x 2 matrix inverse and finding the inverse if it exists. Given an n x n
matrix, determine whether or not the matrix is nonsingular, and if it is,
find its inverse. Understand the basic properties of invertible matrices.
Solve systems of linear equations by matrix inversion. O.
Find the
transpose of a given matrix and demonstrate knowledge of the properties of
the transpose operation. P.
Nature of
solutions of linear equations with special attention to homogeneous systems.
Determine the rank of a given matrix. Q.
Apply
matrices to real life problems including but not limited to cryptography R.
Understand
the definition of a determinant. State the minors and cofactors of a matrix.
Explain when a matrix is in triangular form. Evaluate determinant of special
matrices by inspection. Evaluate the determinant using cofactor expansion. S.
Apply the
determinant test for invertibility. Know the
fundamental properties of the determinant function and its relationship to
inverses. T.
Use the basic
properties of determinants as a short cut in computing the determinant. Evaluate
determinants by row reduction. U.
Apply
Cramer’s Rule to find the solutions of linear systems of n equations in n
unknowns. V.
Distinguish
between the three types of vectors, physical, geometric, and algebraic. Understand vector notation, the position vector
and determine its components, apply the equality test for geometric vectors,
and ability to move geometric vectors about in the plane. W. Understand algebraic vector notation and understand
addition of the three vector types, physical, geometric, and using algebraic
vectors to calculate the sum X.
Understand
multiplying a geometric vector by a scalar multiple and use algebraic vectors
to calculate the scalar multiple of a geometric vector. Understand
subtraction of geometric vectors. Determine parallel vectors, the norm of a
vector, and a standard unit vector in the direction of a given vector.
Understand and apply the basic vectors i & j. Y.
Demonstrate
ability to graph vectors in R3 and understand the righthand
rule. Determine the distance between two points and the midpoint of a line
segment in R3. Z.
Apply vector
concepts covered to the three- dimensional space: geometric, algebraic,
position vector and its components, addition, subtraction, scalar
multiplication, parallel, norm unit and basic vector concepts. AA.
Define the
dot product of two vectors. Be able to compute dot products from vector
components. Apply the algebraic properties of dot product. Determine if two
vectors are orthogonal. Express the sum of two orthogonal vectors. BB.
Define cross
product. Be able to calculate the cross product of two vectors. Know the
basic relationships involving cross and dot products. Understand the
algebraic and geometric properties of the cross product. Know and apply the
tests of collinearity and coplanarity.
Give the geometric interpretation of determinants. CC.
Be able to
use vectors to derive equations of lines and planes in 3-space. Understand
the basic properties of vectors in Rn. DD.
Understand
the definition of eigenvalues and eigenvectors, the relationship to
invertible matrices, and the two-step process for solving the eigenvalue
problem. Use determinants to find eigenvalues. Write the characteristic
polynomial and characteristic equation. Calculate eigenvalues and the corresponding
eigenvectors for a given matrix. EE.
Review and
master complex number concepts. Find the magnitude of a complex number. FF.
Apply complex
number concepts to calculate the eigenvalues and associated eigenvectors of a
matrix having complex eigenvalues. Know and apply the concepts of eigenvalues
to the special matrices: symmetric, triangular, invertible, and transpose.
Use determinants to find eigenvalues. GG.
Use
Gauss-Jordon to solve complex systems. HH.
Understand
the application utilizing eigenvalues involving a difference equation; Markov
chain. |
VI. Test, quizzes, written assignments |

** **VII. **This course
supports the following objectives**:

**DCC Educational Objectives:**

Critical
Thinking

Information Literacy

Quantitative Reasoning

Scientific Reasoning