Introduction to Linear AlgebraMTH 263
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.
- 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.
- 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.
- 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.