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Introduction to Linear Algebra

MTH 263

Introduction to Linear Algebra

MTH 263

Course Description

Prerequisite: MTH 261 with a grade of "C" or better. Investigates matrices, determinants, linear systems, vector spaces, linear transformations, eigenvalues, and eigenvectors. (45-0)

Outcomes and Objectives

The student will learn the fundamental properties of matrices.

Objectives:

  • Add, subtract, multiply, and invert matrices on appropriate occasions and in an appropriate manner.
  • Describe and solve systems of linear equations with matrices.
  • Perform elementary row operations with and without elementary matrices.
  • Define and apply symmetric and skew-symmetric matrices.
  • Define and apply the determinant of a matrix, and the applications of determinants in a variety of contexts.
  • Define and apply eigenvalues and eigenvectors.
  • Recognize digonalizable matrices and transform such matrices into diagonal matrices.
  • Use the specific properties of symmetric matrices.
  • Perform orthogonal diagonalization on symmetric matrices.

The student will learn the fundamental language and processes of vector spaces and inner product spaces.

Objectives:

  • Motivate and execute the definitions of vector space and inner-product space.
  • Recognize vectors, vector spaces, inner-product spaces, and subspaces in a variety of contexts.
  • Define and apply length and orthogonality in a variety of inner-product spaces.
  • Define and apply linear dependence/independence and spanning.
  • Define and apply basis, dimension, and coordinates relative to a basis.
  • Define and apply an orthonormal basis.
  • Define and apply the Gram-Schmidt process.

The student will learn about linear transformations.

Objectives:

  • Motivate and execute the definition of linear transformation.
  • Use the language of linear transformations correctly.
  • Recognize the consequences of linear transformation on dimensions and bases of vector spaces.
  • Identify matrices with linear transformations and to represent linear transformations with matrices.
  • Define and apply the similarity of transformations/matrices.
  • `Employ transition matrices to effect a change of basis.