---
tags:
  - mathematics
  - linear-algebra
  - vectors
  - matrices
created: 2026-05-16
breadcrumb: "Mathematics > Linear algebra > Linear algebra essentials"
---

# Linear algebra essentials

> **Mechanism:** Vectors, matrices, and linear maps

## Items

- **Vector** — An ordered list of numbers; an arrow with magnitude and direction.
- **Vector space** — Set closed under addition and scalar multiplication.
- **Inner product** — ⟨u, v⟩ = Σ uᵢvᵢ. Defines length and angle.
- **Matrix** — Rectangular array; a representation of a linear map.
- **Matrix product** — (AB)ᵢⱼ = Σ AᵢₖBₖⱼ. Composition of linear maps.
- **Determinant** — det(A) — scaling factor of A's linear map. Zero ⇔ singular.
- **Eigenvalue / eigenvector** — Av = λv. Directions A only stretches, factor λ.
- **Rank** — Dimension of the column space; how many independent directions A spans.
- **SVD** — A = UΣVᵀ. Decomposes any matrix into rotation–scaling–rotation.
- **Norm** — ‖v‖ — vector length. ℓ² = √Σ vᵢ².

## Related

- [[calculus-basics]]
- [[famous-math-equations]]