An element of a vector space that can represent direction, magnitude, or a list of numbers

Commonly written as columns or rows (e.g., [v1, v2, ..., vn])

Geometric (2D/3D): arrows in space with direction and length

Algebraic (nD): ordered tuples representing states, features, or coefficients

Vector addition

Scalar multiplication

Linear combinations

A set of vectors closed under addition and scalar multiplication

Must satisfy axioms (associativity, commutativity of addition, existence of zero vector, etc.)

R^n (real n-dimensional vectors)

Polynomials (vectors as polynomial coefficient lists)

Functions (vectors as functions, with addition/scaling defined pointwise)

A subset that is also a vector space under the same operations

Typical forms

Variables appear only to the first power

No products of variables, no nonlinear functions (e.g., sin, exp) of variables

Expresses a system compactly as A x = b

No solution (inconsistent)

Exactly one solution

Infinitely many solutions

Gaussian elimination

Row-reduction to (reduced) row echelon form

Using inverses (when applicable)

Rectangular arrays representing linear transformations and linear systems

Act on vectors via matrix-vector multiplication

Addition and scalar multiplication

Matrix multiplication

Transpose

Inverse (when it exists)

Identity matrix (I): does not change vectors

Diagonal matrices: scaling along coordinate axes

Symmetric matrices: A = A^T

Orthogonal matrices: Q^T Q = I (preserve lengths and angles)

Triangular matrices: upper/lower triangular (useful in elimination)

A function T that preserves addition and scalar multiplication

Every linear transformation between finite-dimensional spaces can be represented by a matrix (given a basis)

Matrix-vector multiplication is the standard representation

Rotations, reflections, scalings, shears, projections

Transformations map lines through the origin to lines through the origin

The set of all linear combinations of a set of vectors

Describes what vectors can be “built” from given vectors

Independent: no vector is a linear combination of the others

Dependent: at least one vector can be formed from others (redundant information)

A minimal spanning set: spans the space and is linearly independent

Provides coordinates for representing any vector uniquely

Number of vectors in a basis

Measures the degrees of freedom of the space

All vectors A x can produce

Span of the columns of A

All vectors x such that A x = 0

Describes degrees of freedom in homogeneous systems

Span of the rows of A (equivalently column space of A^T)

Vectors y such that y^T A = 0

Rank-nullity theorem

The dimension of the column space (or row space)

Indicates how many independent directions the transformation produces

A x = b is solvable iff b is in the column space of A

Over-determined: more equations than unknowns (may be inconsistent; often use least squares)

Under-determined: more unknowns than equations (typically infinitely many solutions)

Scaling factor of area/volume under the transformation

Indicates orientation change (sign) and invertibility (zero vs nonzero)

det(A) ≠ 0 implies A is invertible

det(AB) = det(A) det(B)

Magnitude: how volumes scale

Zero determinant: collapses space into a lower dimension

Nonzero vector v where A v = λ v

Reveal invariant directions and system behavior

Central in stability, oscillations, and repeated application of transformations

When A = P D P^{-1}

Simplifies powers of matrices and many computations

Symmetric matrices can be orthogonally diagonalized

Real eigenvalues and orthogonal eigenvectors

Measures similarity and defines angles

Used to compute projections and orthogonality

||v|| derived from the inner product

u ⟂ v if u · v = 0

Leads to stable bases and simplified computations

Basis vectors are orthogonal and unit-length

Make coordinates and projections straightforward

Finds the closest vector in a subspace to a given vector

Often uses orthonormal bases or normal equations

Solve A x ≈ b when exact solution doesn’t exist

Minimizes ||A x − b|| (best approximation)

Data fitting, regression, signal denoising, measurement error correction

A = L U for efficient solving of many systems with same A

A = Q R with Q orthogonal

Useful for least squares and numerical stability

A = U Σ V^T

Works for any matrix and exposes rank, conditioning, and best low-rank approximations

Encodes complex relationships as linear systems and transformations

Provides efficient algorithms for large-scale problems

Computer graphics (transformations, projections)

Machine learning (feature spaces, optimization, SVD/PCA)

Physics and engineering (systems, eigenmodes, stability)

Networks and search (graphs, adjacency matrices)

Differential equations and dynamical systems (eigenvalues, diagonalization)