Mathematical Notation
Essential mathematical symbols and conventions used throughout the course. Foundation for reading and writing mathematical expressions.
Course Notes
Comprehensive course materials organized by topic.
Set Theory
Fundamental concepts of sets, operations, and mathematical notation
Sets
Introduction to set theory and basic set operations.
Empty Set
Definition and properties of the empty set.
Union
Understanding set unions and their properties.
Intersection
Understanding set intersections and their properties.
Cartesian Product
Understanding the cartesian product operation on sets.
Motivation for the Abstract
Understanding why we study abstract mathematical concepts.
Subsets
Understanding subsets and their properties in set theory.
Functions & Mappings
Functions, mappings, and their properties
Vector Spaces
Vector spaces, subspaces, linear combinations, and basis
Vector Spaces
Fundamental definition and properties of vector spaces, including operations and their generalizations.
Vector Space Axioms
The fundamental axioms that define a vector space.
Vectors
Introduction to vectors and their basic properties.
Linear Combinations
Understanding linear combinations of vectors and their significance.
Vector Spans
Understanding vector spans and their role in vector spaces.
Linear Dependence
Understanding linear dependence and independence of vectors.
Generating Sets
Understanding generating sets (spanning sets) and their role in defining vector spaces.
Vector Subspaces
Understanding vector subspaces and their properties.
Basis
Understanding basis of vector spaces and their significance.
Dimension of a Vector Space
Understanding the concept of dimension in vector spaces and its fundamental properties.
Applications
Systems of linear equations and practical applications
Course Resources
Webwork Information
Online homework system information and access details. Weekly assignments due Thursdays at 3pm.
Midterm 1 Information
Information, study resources, and details for Midterm 1.
Midterm 2 Information
Information, study resources, and details for Midterm 2.
Final Exam Information
Information and details about the final exam.
Linear Transformations
Linear Transformations
Understanding linear transformations between vector spaces and their fundamental properties.
Null Space and Kernel
Understanding null spaces (kernels) of linear transformations and the rank-nullity theorem.
Image of a Linear Transformation
Understanding the image (range) of linear transformations and its properties as a subspace.
Standard Matrix of a Linear Transformation
Understanding the standard matrix representation of linear transformations and how to construct it.
Rank-Nullity Theorem
Fundamental relationship between dimensions of domain, image, and kernel.
Matrices
Matrices
Introduction to matrices, their properties, and their fundamental role in linear algebra.
Gaussian Elimination
A systematic procedure for solving systems of linear equations by transforming a matrix into row-reduced echelon form (RREF).
Row-Reduced Echelon Form (RREF)
Definition and properties of matrices in Row-Reduced Echelon Form (RREF), a standardized form used in linear algebra.
Weekly Summary
Week 1: Introduction to Linear Algebra
Introduction to fundamental concepts including Set Theory, Functions and Mappings, Linear Systems, and Mathematical Notation.
Week 2: Vector Spaces
Introduction to vector spaces, including vector operations, subspaces, and linear combinations.
Week 3: Linear Independence and Basis
Exploring linear independence, spans, and the concept of basis in vector spaces.
Week 4: Vector Spaces and Linear Dependence
Advanced concepts in Vector Spaces and further exploration of Linear Dependence built on Linear Combinations.
Week 5: Midterm Review
Review
Week 6: The Matrix
Matrices and dat
Week 7: Dimensions and Linear Transformations
Dimensions of vector spaces, linear transformations, and null spaces - fundamental concepts building upon vector spaces.
Week 10: Midterm 2 Review
Review materials and resources for Midterm 2
Week 11: Determinants & Scaling
Understanding determinants and their geometric meaning through scaling
Week 12: Eigenvalues and Eigenvectors
Eigenvalues, eigenvectors, and their role in linear transformations