Primary Textbooks
Introduction to Linear Algebra
Strang, Gilbert. 5th edition. Wellesley-Cambridge Press, 2016. ISBN: 978-0-9802327-7-6.
The standard reference for MIT's 18.06 course. Our notation for RREF, pivot, and elimination steps follows Strang's conventions. Theorems are cited by chapter and section where referenced.
Linear Algebra and Its Applications
Lay, David C., Steven R. Lay, and Judi J. McDonald. 5th edition. Pearson, 2016. ISBN: 978-0-321-98238-4.
Our secondary reference, particularly for application examples in the guides. Lay uses the same RREF conventions as Strang and is widely used in US universities.
Online Courses
MIT OpenCourseWare — 18.06 Linear Algebra
Gilbert Strang, Massachusetts Institute of Technology. Available at ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/.
The lecture videos and problem sets from MIT's flagship linear algebra course are freely available and serve as the pedagogical backbone for many of our explanations.
Schema and Standards
Schema.org MathSolver
schema.org/MathSolver — Used for structured data markup on all calculator pages to signal math-solving functionality to search engines.
KaTeX
Khan Academy's fast math rendering library. Used for all formula and matrix rendering on this site. katex.org.
Mathematical Background
The following facts are standard results in linear algebra. We reference them without proof but can cite the specific theorem in Strang or Lay:
- RREF Uniqueness Theorem: Every matrix has a unique RREF. (Strang, §1.5; Lay, §1.2 Theorem 1)
- Rank-Nullity Theorem: rank(A) + nullity(A) = n. (Strang, §3.3; Lay, §4.6 Theorem 14)
- Invertible Matrix Theorem: A square matrix is invertible iff det ≠ 0 iff rank = n iff RREF = I. (Lay, §2.3 Theorem 8)
- Cofactor Expansion: det(A) = Σⱼ (−1)^(i+j) aᵢⱼ det(Mᵢⱼ). (Strang, §5.2; Lay, §3.1)
- Cross Product as Determinant: u × v = det([i,j,k; u; v]). (Standard multivariable calculus result.)