Linear Algebra
About This Series
Linear algebra is a foundational branch of mathematics dealing with vectors and matrices. This series begins with the basics of vectors and matrices, then progresses through vector spaces, various derivations of determinants, eigenvalues and diagonalization, and finally applications.
Linear algebra underpins all areas of mathematics including analysis, geometry, and abstract algebra, and has become an essential tool in physics, engineering, data science, and machine learning.
Learning by Level
Learning Path
Key Topics
Vector Spaces
Definition of abstract vector spaces, linear independence, bases, and the concept of dimension.
Determinants
Definition and properties of determinants. Multiple approaches including Cramer's rule, cofactor expansion, the Leibniz formula, and exterior algebra.
Eigenvalues and Eigenvectors
Understanding the "essence" of a matrix. Diagonalization, complex eigenvalues, and spectral decomposition.
Applications
Differential equations, Markov chains, principal component analysis (PCA), and Google PageRank.
Frequently Asked Questions
Q1. What is linear algebra?
Linear algebra is a foundational branch of mathematics dealing with vectors and matrices. It systematically covers vector spaces, linear maps, determinants, eigenvalues, and eigenvectors. It serves as the foundation for analysis, geometry, and abstract algebra, and is an essential tool in physics, engineering, data science, and machine learning.
Q2. In what order should I study linear algebra?
This series offers four levels: Introductory (only high-school-level mathematics needed: vectors, matrices, systems of equations) → Elementary (university years 1-2: vector spaces, eigenvalue basics) → Intermediate (university years 2-3: diagonalization, determinant derivations) → Advanced (university year 3 to graduate: rigorous proofs, exterior algebra, applications).