Linear Algebra — Basic
Fundamentals of Vector Spaces, Eigenvalues, and Determinants (Undergraduate Level)
Overview
The basic level covers three core concepts of linear algebra: vector spaces, eigenvalues and eigenvectors, and determinants. These are fundamental tools used throughout modern mathematics.
Learning Objectives
- Understand the abstract definition of vector spaces
- Grasp the concepts of linear independence and basis
- Learn the definition and computation of eigenvalues and eigenvectors
- Master basic determinant computation methods (cofactor expansion, Cramer's rule)
Table of Contents
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Chapter 1
Fundamentals of Vector Spaces
Axiomatic definition, beyond $\mathbb{R}^n$ — polynomials and functions as vectors
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Chapter 2
Linear Independence and Basis
Linear combinations, linear independence, basis, dimension
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Chapter 3
Introduction to Determinants: Cramer's Rule
Deriving the determinant from systems of equations — a historical approach
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Chapter 4
Determinants: Cofactor Expansion
Cofactors, minors, Laplace expansion, inverse matrices
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Chapter 5
Visual Understanding of Determinants
Shear transformations, parallelograms, changes in area and volume
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Chapter 6
Eigenvalues and Eigenvectors
Definition, geometric meaning, characteristic polynomial, eigenspaces
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Chapter 7
Properties and Applications of Eigenvalues
Trace and determinant, triangular matrices, linear independence, applications
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Chapter 8
Structure of Solutions of Linear Systems
Homogeneous and non-homogeneous systems, solution space structure theorem, rank-nullity theorem
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Chapter 9
The Identity Matrix
Definition and basic properties, eigenvalues, the identity element of matrix multiplication
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Chapter 10
The Kronecker Delta
Definition of δ_ij, Einstein notation, foundations of tensor notation
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Chapter 11
Matrix Rank
Definition of rank, pivot-based computation via row reduction, rank-nullity theorem, full-rank equivalences. Appendix: Strang-style proof of row rank = column rank
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Chapter 12
Permutation Matrices
Definition, orthogonality, determinant, relation to LU decomposition, symmetric group, applications
Prerequisites
- High school-level vectors (arrow vectors, component representation)
- Basic matrix operations (addition, multiplication, inverse)
- Fundamental concepts of sets and mappings