# Linear Algebra

Linear algebra is the branch of mathematics concerning **vector spaces**, **linear transformations**, and **systems of linear equations**. It is foundational to nearly all areas of pure and applied mathematics, physics, engineering, computer science, and data science.

## Goals

- **Unify concepts** — Connect systems of equations, matrices, vector spaces, and linear transformations under a single conceptual framework
- **Build intuition** — Develop geometric and algebraic understanding of abstract structures
- **Provide rigor** — Formal definitions, theorems, and proofs for key results
- **Show applications** — Demonstrate relevance to differential equations, Markov chains, cryptography, coding theory, and more

## Features

- **Systems of linear equations** — Gaussian elimination, echelon forms, parametric solution sets
- **Matrix theory** — Operations, inverses, determinants, factorizations (LU, QR, SVD), canonical forms
- **Vector spaces** — Subspaces, bases, dimension, coordinate changes, abstract vector spaces
- **Linear transformations** — Kernel, range, isomorphisms, matrix representations, eigenvalues
- **Eigenvalues & eigenvectors** — Characteristic polynomials, diagonalization, Jordan form, spectral theory
- **Orthogonality** — Inner products, projections, Gram-Schmidt, orthogonal matrices
- **Applications** — Differential equations, Markov chains, error-correcting codes, input-output analysis

## Structure

This topic is organized into five main branches:

1. **Linear systems of equations** — The starting point: solving Ax = b, consistency, echelon forms
2. **Matrices** — The central computational tool: operations, properties, decompositions
3. **Linear systems and matrices** — The bridge: connecting systems to matrix equations
4. **Vector spaces** — The abstract framework: R^n, subspaces, bases, dimension, isomorphisms
5. **Applications** — Real-world uses: differential equations, Markov chains, coding theory

## Open Questions

- How to best visualize high-dimensional vector spaces?
- Connections to abstract algebra (modules over rings)
- Numerical linear algebra and computational complexity