Linear Algebra Cheat Sheet
Your complete reference for first-year university linear algebra. From vectors and matrices to eigenvalues and transformations — all the formulas and concepts you need to succeed.
Vectors & Spaces
5 topics
Introduction to Vectors
Vector notation, geometric interpretation, column and row vectors, and basic terminology.
Vector Operations
Vector addition, scalar multiplication, dot product, cross product, and their properties.
Vector Spaces
Definition of vector spaces, axioms, examples including Rⁿ, and closure properties.
Subspaces
Subspace definition, subspace test, null space, column space, and row space.
Basis and Dimension
Linear independence, spanning sets, basis vectors, and dimension of vector spaces.
Matrices
7 topics
Matrix Basics
Matrix notation, types of matrices (square, diagonal, identity, zero), and basic terminology.
Matrix Addition & Scalar Multiplication
Adding matrices, scalar multiplication, properties, and the zero matrix.
Matrix Multiplication
Matrix product definition, properties, non-commutativity, and applications.
Special Matrices
Symmetric, skew-symmetric, orthogonal, triangular, and diagonal matrices.
Matrix Transpose
Transpose definition, properties, transpose of products, and symmetric matrices.
Inverse Matrices
Invertible matrices, finding inverses, properties of inverse, and singular matrices.
Matrix Rank
Definition of rank, rank-nullity theorem, and applications to linear systems.
Systems of Equations
4 topics
Systems of Linear Equations
Representing systems as matrices, Ax = b form, and geometric interpretation.
Gaussian Elimination
Elementary row operations, forward elimination, and back substitution.
Row Echelon Form
REF and RREF, pivot positions, leading ones, and reduction algorithm.
Solution Types
Consistent vs inconsistent systems, unique vs infinite solutions, and free variables.
Determinants
4 topics
Determinant Basics
Definition for 2×2 and 3×3 matrices, geometric interpretation as area/volume.
Properties of Determinants
Row operations effects, multiplicative property, transpose, and invertibility.
Cofactor Expansion
Minors, cofactors, Laplace expansion along rows and columns.
Cramer's Rule
Solving linear systems using determinants, applications and limitations.
Eigenvalues & Eigenvectors
5 topics
Eigenvalue Basics
Definition of eigenvalues and eigenvectors, the equation Av = λv, and geometric meaning.
Characteristic Polynomial
Finding eigenvalues using det(A - λI) = 0, characteristic equation.
Finding Eigenvectors
Solving (A - λI)v = 0, eigenspaces, and algebraic vs geometric multiplicity.
Diagonalization
Diagonalizable matrices, P⁻¹AP = D, conditions for diagonalization.
Applications of Eigenvalues
Matrix powers, differential equations, Markov chains, and principal component analysis.
Linear Transformations
5 topics
Introduction to Linear Transformations
Definition, examples (rotation, scaling, projection), and linearity conditions.
Kernel and Range
Null space as kernel, column space as range, and dimension theorem.
Matrix Representation
Representing transformations as matrices, standard matrix, and change of basis.
Composition of Transformations
Composing transformations, matrix multiplication interpretation, and order matters.
Orthogonal Transformations
Orthogonal matrices, rotations, reflections, and preserved distances.
Ready to Master Linear Algebra?
Bookmark this page and use it as your go-to reference for all linear algebra formulas and concepts. Each topic includes detailed explanations and worked examples.