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.  Learner Outcomes

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.  Evaluation

 

Test, quizzes, written assignments

 VII.    This course supports the following objectives:

DCC Educational Objectives:

Critical Thinking
Information Literacy
Quantitative Reasoning
Scientific Reasoning