# 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.