School

Science, Technology, Engineering and Math

Division

Mathematics

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 non-graphing scientific calculator for some formal assessments, access to a free graphing utility application to support learning and during informal assessments, and access to an online homework assessment.

## 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 B’ 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 final 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 final exam may be a common final exam administered to all sections of Math-283.
- All students will be required to complete at least two proctored on-campus exams. The cumulative value of those exams must be at least 40% of a student’s final grade.
- All students will be required to complete online homework. This online homework must be weighted in such a manner so that it is worth between six percent (6%) and twelve percent (12%) of the final course grade.
- Additional assessment of student achievement may include assignments, quizzes, and exams.
- For proctored in-person formal assessments (quizzes, tests, and exams) the only technology students can use is a non-graphing scientific calculator. Quizzes, tests, and exams may have non-calculator parts. Class projects and informal assessments will require students to use a free graphing application to support learning.
- 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 calculator.

General Course Requirements and Recommendations

- A non-graphing scientific calculator is required for proctored in-person formal assessments.
- Free graphing applications will be used to support learning, informal assessments, and class work.
- Access to an online homework management system is also required.
- Application problems must be covered in all mathematics courses. Every section in any course outline that includes application problems must be covered.

## Outcomes

General Education Categories

- Mathematics

MTA Categories

- Category 3: Mathematics (College Algebra Track)

Satisfies Wellness Requirement

No

## 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.

Effective Term

Fall 2024