# Course Notes: Linear Algebra – MATH 2130

This course is an introduction to linear algebra. Topics we will cover include basic properties of systems of linear equations, matrices and matrix algebra, determinants, vector spaces, subspaces, linear independence of vectors, basis and dimension of subspaces, linear transformations, eigenvalues and eigenvectors of a matrix, orthogonality of vectors, inner product and length of vectors.

## Final Review – MATH 2130

Gram-Schmidt Process

## Gram-Schmidt – MATH 2130

Gram-Schmidt Process

## Orthogonal Projection – MATH 2130

Orthogonal Projection and Orthogonal Complement

Orthogonal Sets

## Inner Product – MATH 2130

The euclidean inner product

## Diagonalization – II – MATH 2130

Diagonalization – Part II

## Diagonalization – I – MATH 2130

Diagonalization – Part I

## Characteristic Equation – MATH 2130

Characteristic Equation

## Eigenvalues and Eigenvectors – MATH 2130

Eigenvalues and eigenvectors

## Midterm II – Review II – MATH 2130

Second review for Midterm II

## Midterm II – Review I – MATH 2130

First review for Midterm II

## Change of Basis – Examples – MATH 2130

Changing basis in vector spaces

## Change of Basis – MATH 2130

Changing basis in vector spaces

## Dimension of general vector spaces – MATH 2130

General definition of dimension

## Properties of the Coordinate Map – MATH 2130

Properties of the Coordinate Map

## Coordinate systems for general vector spaces – Part I- MATH 2130

Coordinate vectors, Coordinate Mapping for general vector spaces. Part I

## Linear (in)dependence and Bases for general vector spaces – MATH 2130

Linear (in)dependence and Bases for general vector spaces

## kernel and Range of linear transformations – MATH 2130

Kernel and Range of linear transformations

## Examples of vector spaces – MATH 2130

Examples of vector spaces in Computer Science

## (Updated) Abstract vector spaces – MATH 2130

Abstract definition of vector spaces, subspaces, linear (in)dependence and examples

## Cramer’s rule, areas and volumes – MATH 2130

Cramer’s Rule, formula for inverse of a matrix and the geometric description of determinants

## Useful properties of determinants – MATH 2130

Some properties of determinants that can useful for computations

## Introduction to determinants – MATH 2130

Introduction to Determinants

## Dimension and Rank – MATH 2130

Dimension of spaces and Rank of a matrix

## Review II – Midterm I – MATH 2130

Review II for Midterm I exam

## Review I – Midterm I – MATH 2130

Review I for Midterm I exam

Subspaces of R^n

## The Inverse of a Matrix – MATH 2130

Inverse of a Matrix

## Matrix Operations – MATH 2130

Sum and multiplication of matrices

## The Standard Matrix of a Linear Transformation – MATH 2130

The standard Matrix of a Linear Transformation

## Introduction to Linear Transformations – MATH 2130

Introduction to Linear Transformations

## Linear (in)dependence – MATH 2130

Introduction to linear (in)dependence

## Solution sets of matrix equations – MATH 2130

Describing solution sets in the parametric vector form

## The Matrix Equation AX=B – MATH 2130

A new way to see linear systems: matrices acting on vectors.

## Introduction to vectors – MATH 2130

Introduction to vectors

## Describing Solutions of a Linear System – MATH 2130

Describing solutions of a Linear System in the parametric way.

## The Row Reduction Algorithm – MATH 2130

The Algorithm to obtain the reduced echelon form of a matrix. Useful to solve linear systems.

## Could there be a linear system with exactly two solutions?

Proof that if a linear system has two distinct solutions, then it has infinitely many solutions.

## (Reduced) Echelon Form – MATH 2130

Echelon Forms of Matrices

## The Matrix Map – MATH 2130

Encoding a linear system into a matrix

## Solving Linear Systems – Part I – MATH 2130

Defining the basic rules to solve linear systems

## First definitions – MATH 2130

Presentation covering the basic definitions of the course.