Linear operator

A map between vector spaces that preserves vector addition and scalar multiplication. In finite dimensions, it can be represented by a matrix after choosing bases.

Vector space: A set equipped with vector addition and scalar multiplication satisfying specified algebraic axioms. Linear operators take vector spaces as their domains and codomains.
Matrix representation: A matrix that encodes a linear map relative to chosen bases of its domain and codomain. It makes operator calculations explicit once coordinate systems are fixed.
Rotation matrix: A matrix representing a rotation of Euclidean space while preserving lengths and angles. It is a familiar geometric example of a linear operator.
Affine transformation: A map formed by a linear transformation followed by a translation. Translations generally break linearity by moving the zero vector.
Linear transformation: A function between vector spaces that preserves addition and scalar multiplication. This is the general name for the structure-preserving maps called operators.
Composition of linear maps: The operation of applying one linear map after another when their domains and codomains match. Composing operators corresponds to multiplying their matrix representations.
Differential operator: An operator built from differentiation that acts on functions. Differentiation provides a central operator on spaces of functions.
Nonlinear operator: A map between function spaces that fails to preserve linear combinations. It lacks the defining additivity or scalar compatibility of a linear operator.
Basis (linear algebra): A linearly independent set of vectors that spans a vector space. Choosing bases turns a linear operator into a matrix representation.
Identity operator: The linear operator that maps every vector to itself. It serves as the neutral element for composing operators.

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