Linear Algebra
About This Series
Linear algebra is a foundational branch of mathematics dealing with vectors and matrices. This series begins with the concept of vector spaces, then progresses through 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
Introductory
High school level
- Basics of vectors and matrices
- Matrix multiplication, systems of equations
- Inverse matrices and determinants
Advanced
University year 3 to graduate
- Rigorous proof of the Leibniz formula
- Axiomatic definition and uniqueness
- Exterior algebra
- Applications of eigenvalues (PCA, PageRank)
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.