# MATH 225: Introduction to Linear Algebra

## Description

The course introduces systems of linear equations and techniques of finding their solutions based on row operations and Gaussian elimination. Fundamentals of matrix algebra including matrix product and matrix inverses and their properties are studied.

Linear combinations and independence, subspaces, basis, and dimension are treated in the context of Rn, culminating in the fundamental matrix subspaces (range and kernel) and rank-nullity theorem. These will be studied to enhance students’ geometric intuition about linear algebra in R2 and R3. Linear transformations are introduced as matrix transformations from Rn to Rm together with change of basis with an eye towards diagonalization. Eigenvalues and eigenvectors are covered after a light introduction to determinants. Orthogonality is treated in the context of the dot product leading to orthogonal bases, orthogonal projections, Gram-Schmidt orthogonalization, and their applications. The course also illustrates techniques of linear algebra via software such as MATLAB.

## Prerequisite

Precalculus (Math 199 or equivalent);

## Content

1. Solution of system of linear equations using Gaussian elimination
2. Matrix algebra, special matrices (diagonal, triangular, symmetric), matrix inverse and transpose
3. Linear dependence/independence, subspaces in Rn, their bases and dimension (rank)
4. Range and kernel of a matrix, finding their bases, rank-nullity theorem
5. Linear transformations as matrix transformations from Rn to Rm and change of basis
6. Determinants (light)
7. Eigenvalues, eigenvectors, eigenspaces and their bases
8. Diagonalization and applications
9. Euclidean inner product (dot product) and norm of vectors in Rn
10. Cauchy-Schwarz and triangle inequalities (time permitting)
11. Orthogonality, orthogonal bases, orthogonal projections
12. Gram-Schmidt orthogonalization (time permitting), least-squares solutions

## Student Learning Outcomes

Upon successful completion of the course, the students will be able to :

1. Find solutions of systems of linear equations using Gaussian elimination
2. Use bases and orthonormal bases to solve problems in linear algebra
3. Find the dimension of subspaces such as those associated with matrices
4. Find eigenvalues and eigenvectors and use them in applications

## Suggested textbooks

1. Linear Algebra with Applications, W. Keith Nicholson, (Lyryx Open Source)
2. Linear Algebra and its Applications (5th ed.), David C. Lay, Steven R. Lay, Judi J. McDonald

Date: January 23, 2023