Linear map

A function between vector spaces that preserves vector addition and scalar multiplication. These properties make it compatible with linear combinations.

Vector space: A set with vector addition and scalar multiplication satisfying the vector-space axioms. Linear maps are defined between vector spaces and preserve their two operations.
Matrix representation: A matrix encoding a linear map after bases are chosen for its domain and codomain. Its columns are the map's outputs on the domain basis vectors.
Linear transformation: A linear map from a vector space to itself, often viewed as a transformation of that space. It is the endomorphism case, where inputs and outputs belong to the same space.
Affine transformation: A map formed by a linear transformation followed by a translation. A nonzero translation breaks preservation of vector addition, unlike a linear map.
Linear combination: A sum of scalar multiples of vectors. Preserving addition and scalar multiplication means preserving every linear combination.
Matrix multiplication: An operation that combines matrices according to row-by-column products. Multiplying representing matrices computes the composition of linear maps.
System of linear equations: A collection of equations whose variables occur only to the first power and are not multiplied together. A matrix equation packages such a system as a linear map applied to an unknown vector.
Nonlinear function: A function that does not satisfy the defining linearity conditions. Failure of additivity or scalar compatibility separates it from a linear map.
Basis: A linearly independent set of vectors that spans a vector space. A linear map is determined completely by its values on a basis.
Composition of functions: A function formed by applying one function to the output of another. Composing linear maps produces another linear map, represented by matrix multiplication.

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