I. Introduction

Linear Algebra is the math of vectors and vector spaces. It explains how linear systems, matrices, and transformations work, and it provides the core language for geometry and computation in any number of dimensions.

II. Outline

  • Linear Algebra I - Foundations & Systems
    • Vectors, linear combinations, span
    • Systems of linear equations (Gaussian elimination)
    • Matrices and matrix algebra
    • Linear independence, basis, dimension
    • Column space / row space, rank, null space (and the rank–nullity idea)
    • Determinants
  • Linear Algebra II - Linear Transformations & Eigenstuff
    • Linear transformations, kernels/images, change of basis
    • Eigenvalues & eigenvectors
    • Diagonalization, Jordan form
    • Complex eigenvalues, stability intuition
    • Applications: decoupling systems, Markov chains, graph ideas
  • Inner Product Spaces - Geometry & Orthogonality
    • Dot product, norms, angles, projections
    • Orthogonality, orthonormal bases
    • Gram–Schmidt process
    • Least squares & solving “overdetermined” systems
    • Orthogonal matrices and geometric meaning
  • Advanced / Applied Linear Algebra
    • Singular Value Decomposition (SVD) and the “best approximation” idea
    • PCA and data compression intuition
    • Spectral theorem (symmetric/Hermitian matrices)
    • Numerical linear algebra basics (conditioning, stability)
    • Applications: signals, graphics, control, ML, networks

III. Free Books

IV. Video Series

V. See Also

Systems of Equations, Matrices, Determinants, Eigenvalues/Eigenvectors, Differential Equations, Multivariable Calculus, Numerical Methods