School

Science, Technology, Engineering & Math

Department

Mathematics

Academic Level

Undergraduate

Course Subject

Mathematics

Course Number

283

Course Title

Linear Algebra

Credit Hours

3.00

Instructor Contact Hours Per Semester

47.00 (for 15-week classes)

Student Contact Hours Per Semester

47.00 (for 15-week classes)

Grading Method

A-E

Pre-requisites

MATH-183 with a C or better

Catalog Course Description

Introduces matrix and linear algebra. Covers systems of linear equations, matrix operations and properties of matrices, determinants, the n-dimensional real vector space, general vector spaces, inner product spaces, linear transformations, and eigenvalues and eigenvectors. Presents various applications. Requires a graphing calculator with the TI-84 Plus series recommended.

### Goals, Topics, and Objectives

Goal Statement

- To provide an introduction to the concepts and techniques of linear algebra and matrix algebra for students of engineering, mathematics, the physical sciences, and the social sciences.
- To illustrate some of the widespread applications of matrix algebra and linear algebra.
- To demonstrate the axiomatic method in the context of matrix theory and vector space theory.

Core Course Topics

- Systems of Linear Equations
- Solve a system of linear equations using Gaussian elimination or Gaussian elimination with back-substitution.
- Solve a homogeneous system of linear equations.
- Determine whether a matrix is in row-echelon or reduced-row-echelon form.
- Use elementary row operations on an augmented matrix, perhaps combined with back-substitution, to solve a system of linear equations.
- Set up and solve a system of equations to fit a polynomial function to a set of data points.
- Write and solve a system of linear equations in the form Ax = b.

- Matrices
- Use properties of matrix operations to solve matrix equations.
- Find the transpose of a matrix, the inverse of a matrix, and the inverse of a matrix product (if they exist).
- Factor a matrix into a product of elementary matrices.
- Complete proofs involving matrix properties and operations.
- Solve an application problem involving stochastic matrices, cryptography, input-output models, or least-squares regression (two of these four).

- Determinants
- Use expansion by cofactors to find the determinant of a matrix.
- Use elementary row or column operations to evaluate the determinant of a matrix.
- State conditions that yield zero determinants.
- Find the determinants of elementary and triangular matrices.
- Use the determinant to determine whether a matrix is singular or nonsingular.
- Recognize equivalent conditions for a nonsingular matrix.
- Complete proofs involving determinant properties and operations.

- Vector Spaces
- Perform vector operations on vectors in R^n.
- Prove or disprove that a given set of vectors with two operations is a vector space.
- Prove or disprove that a given subset W of a vector space V is a subspace of V.
- Write a linear combination of a finite set of vectors from a vector space V.
- Determine whether a set S of vectors in a vector space V spans V.
- Determine whether a set S of vectors in a vector space V is linearly independent.
- Determine whether a set S of vectors in a vector space V is a basis for V.
- State standard bases for the vector spaces R^n, Mm,n, and Pn.
- Find a basis for, and the dimension of, the column space or row space of a matrix.
- Find a basis for, and the dimension of, the nullspace of a matrix.
- Express the general solution of a consistent system Ax = b in the form xp + xh.
- Find the transition matrix from a basis B to a different basis Bâ€™ in R^n.
- Find the coordinate vector for a vector in a vector space V with respect to a nonstandard basis.
- Apply the concept of linear independence to find the general solution of a differential equation or apply the concept of a transition matrix to perform a rotation of axes to graph a conic section (one of these two).

- Inner Product Spaces
- Find the length of a vector and a unit vector in the same direction as (or in the opposite direction from) a vector.
- Given two vectors, find the distance between them, their dot product, and the angle between them.
- Determine whether two vectors are orthogonal, parallel, or neither.
- Prove or disprove that a given function defines an inner product on a vector space.
- Find the inner product of two vectors.
- Find the projection of a vector onto a vector or subspace.
- Determine whether a set of vectors is orthogonal, orthonormal, or neither.
- Find the coordinates of a vector relative to an orthonormal basis.
- Use the Gram-Schmidt orthonormalization process.
- Find an orthonormal basis for the solution space of a homogeneous system.

- Linear Transformations
- Prove or disprove that a given function from one vector space to another is a linear transformation.
- Find the image and preimage of a vector under a linear transformation.
- Find the kernel, the range, and the bases for the kernel and the range of a linear transformation T, and determine the nullity and rank of T.
- Determine whether a linear transformation is one-to-one or onto.
- Determine whether two vector spaces are isomorphic.
- Find the standard matrix for a linear transformation.
- Use the standard matrix for a linear transformation to find the image of a vector.
- Find the standard matrix of a composition of two linear transformations.
- Determine whether a linear transformation is invertible, and find its inverse if it exists.
- Find the matrix of a linear transformation relative to a nonstandard basis.
- Use the definition and properties of similar matrices.
- Identify linear transformations defined by reflections, expansions, contractions, shears, and/or rotations.

- Eigenvalues and Eigenvectors
- Find the characteristic equation and the eigenvalues and corresponding eigenvectors of a matrix A.
- Determine whether a matrix is diagonalizable or orthogonal.
- Find a nonsingular matrix P that diagonalizes a matrix A (if possible).
- Find, if possible, a basis B for the domain of a linear transformation T such that the matrix of T relative to B is diagonal.
- Find the eigenvalues of a symmetric matrix and determine the dimension of the corresponding eigenspace.
- Find an orthogonal matrix P that diagonalizes A.
- Use an age transition matrix to solve a population problem, solve a system of first-order linear differential equations, find a matrix of the quadratic form associated with a quadratic equation, or perform a rotation of axes to rotate a quadric surface (two of these four).

### Assessment and Requirements

Assessment of Academic Achievement

- All students will be required to complete a comprehensive final examination that assesses the learning of all course objectives. This exam must be weighted in a manner so that this exam score is worth a minimum of fifteen percent (15%) of the final course grade. In selected semesters this exam may be a common exam administered to all sections of Math 283.
- Additional assessment of student achievement may include assignments, quizzes, and exams.
- Application problems must not only be included on chapter exams but also on the final exam.
- Some exam problems should require the use of a graphing calculator.

General Course Requirements and Recommendations

- Application problems must be covered in all mathematics courses. Every section in any course outline that includes application problems must be covered.
- A graphing calculator is required of each student. The Mathematics Department recommends and uses the TI-84 Plus series.

### Credit for Prior College-Level Learning

Other Exam Details

A student may receive credit for MATH-283 by earning at least 5 on the International Baccalaureate-Higher Level (IB-HL) Further Mathematics Exam.

### Approval Dates

Effective Term

Fall 2020