Developmental

• Math 372
• Math 372L
• Math 376
• Math 376L
• Math 380
• Math 380L
• Math 101
• Math 120
• Math 120L

College Transfer

• Math 4
• Math 5
• Math 15
• Math 25
• Math 30
• Math 45
• Math 50A
• Math 50B
• Math 50C
• Math 52
• Math 55

Course Proposals (Drafts)

Course Outlines

• Course Outline Home Page

Math 45 --- Linear Algebra

Click here to download Curriculum Committee document

Catalog Description

The use and application of matrices in the solution of systems of linear equations, determinants, vector spaces, linear transformations, eigenvalues, eigenvectors, diagonalization, and orthogonality. Linear algebra is a core course in many engineering, physics, mathematics, and computer science programs.

Special notes or advisories: Computer exploration is an integral component of this course. Students will also create and present oral and written analyses of a topic that requires use of the concepts and techniques learned in this course.

Prerequisites

Math 30 (or equivalent) and Math 25 (or equivalent) with a grade of "C" or better, or appropriate score on assessment exam.

Describe representative skills without which the student would be highly unlikely to succeed: Students must be well-grounded in both college algebra and trigonometry to be successful in this course.

Recommended Preparation

Course Learning Outcomes

What should the student be able to do as a result of taking this course? State some of the objectives in terms of specific, measurable student accomplishments.

1. Read, write, and speak accurately about mathematical ideas and use correct mathematical notation.
2. Use computer technology to perform matrix computations, explore mathematical concepts, and verify their work.
3. Use the theory of matrices as a fundamental problem-solving tool.
4. Perform matrix computations, solve linear systems of equations and determine the bases of related vector spaces, find eigenvalues and eigenvectors, and diagonalize matrices when applicable.
5. Apply the mathematics of linear algebra to real-world problems and applications.
6. Read and write mathematical proofs.
7. Use numerical, graphical, symbolic, and verbal representations to solve problems and communicate with others.

Course Content

• Themes: What themes, if any, are threaded throughout the learning experiences in this course?
  • Problem solving
  • Writing
  • Technology
  • Communication
  • Logic
• Concepts: What concepts do students need to understand to demonstrate course outcomes?
  • The use of mathematical software as a fundamental problem-solving tool, particularly in solving linear systems of equations and determining the bases of related vector spaces.
  • The presentation of mathematical solutions, including proofs, in a logical coherent structure, including the use of fundamental writing skills, grammar, and punctuation.
  • Linear systems of equations.
  • Matrices.
  • Vector spaces.
  • Linear transformations.
  • Linear independence, span, and basis.
  • Eigenvalues and eigenvectors.
  • Diagonalization.
  • Orthogonality.
• Issues: What primary issues or problems, if any, must students understand to achieve course outcomes.
  • The appropriate use of technology in the problem-solving process.
  • The connection between mathematics and the real world.
  • The role of the student in becoming a successful learner.
  • The recognition that the problem-solving skills learned in this class are applicable in future mathematics classes and classes in related fields, such as physics, engineering, etc.
  • Pitfalls of using numerical approximations.
• Skills: What skills must students master to demonstrate course outcomes?
  1. Systems of linear equations:
     • Use matrix methods to solve systems of linear equations, and interpret the results geometrically in two and three dimensions.
     • Express solutions in parametric form, and interpret the results geometrically in two and three dimensions.
     • Apply linear algebra theory to determine the existence and uniqueness of solutions of particular systems.
     • Use systems of linear equations to solve mathematical problems and application problems.
  2. Matrix algebra and determinants:
     • Perform matrix operations.
     • Compute the inverse of a matrix using row reduction.
     • Compute the determinant of a matrix.
     • Use Cramer's rule to solve a linear system of equations.
     • Use matrix algebra to solve mathematical problems and application problems.
  3. Vector spaces:
     • Determine if a set of vectors spans a given vector space, and interpret geometrically.
     • Determine if a set of vectors is linearly independent, and interpret geometrically.
     • Find bases for the nullspace, column space, and row space of a matrix.
     • Compute the rank and nullity of a matrix.
     • Reduce a spanning set to form a basis.
     • Expand a linearly independent set to form a basis.
  4. Linear transformations:
     • Analyze the action of a 2x2 matrix as a function and determine its effect on objects in two dimensions.
     • Find the matrix of a linear transformation relative to given bases.
     • Find bases for the nullspace and range of a linear transformation.
     • Find the matrix which implements a change of basis, and use it to compute the matrix of a linear transformation relative to the new basis.
     • Use linear transformations and determinants to solve geometrical problems and application problems.
  5. Eigenvalues and eigenvectors, diagonalization:
     • Develop a geometrical interpretation of the definition of an eigenvalue and eigenvector.
     • Find the eigenvalues of a matrix.
     • Find a basis for each eigenspace of a matrix.
     • Find a basis which diagonalizes a given matrix, or determine that the matrix is not diagonalizable.
     • Apply the theory of eigenvalues, eigenvectors, and diagonalization to solve mathematical problems and application problems.
  6. Orthogonality:
     • Compute inner products, find the length of a vector, and find the angle between vectors.
     • Compute the orthogonal projection of a vector onto a subspace.
     • Use an orthogonal projection to compute the least-squares solution of Ax=b.
     • Transform a given basis into an orthonormal basis.
     • Use orthogonal matrices to diagonalize a given symmetric matrix.

Representative Learning Activities

What will the students be doing (i.e., Listening to lectures, participating in discussions and/or group activities, attending a field trip, etc.)? Relate the activities directly to the Course Learning Outcomes.

• Listening to lectures.
• Participating in group activities and/or assignments.
• Participating in class assignments and/or discussions.
• Completing homework assignments.
• Using mathematical software to complete activities designed to foster a deeper level of understanding of the concepts and skills developed in this class.

Assessment Tasks

How will the student show evidence of achieving the Course Learning Outcomes? Indicate which assessments (if any) are required for all sections.

• Representative assessment tasks:
  • Homework assignments.
  • In-class examinations and/or quizzes.
  • Take-home examinations and/or quizzes allow the instructor to include questions and/or exercises that require the use of a computer and mathematical software to supplement the analysis. Extra time allows the students to develop their writing and presentation skills.
  • Writing assignments designed to develop communication of mathematical concepts.
  • Group and/or individual projects and presentations.
• Required assessments for all sections – to include but not limited to:
  • Homework assignments, including computer activities that deepen the level of understanding.
  • Writing assignments designed to develop communication of mathematical concepts.
  • Student projects
  • Examinations.

Examples of Appropriate Texts or Other Readings

• Author: Strang; Title: Introduction to Linear Algebra, Third Edition; Date: 2003
• Author: Lay; Title: Linear Algebra and Its Applications, Third Edition Update; Date: 2006