1.2.13 Linear Map Definition

Linear Map Definition is the characterization of a linear map as a function between two vector spaces, defined over the same scalar field, that preserves the operations of vector addition and scalar multiplication, meaning that the image of a sum equals the sum of the images and the image of a scaled vector equals the scaling of the image. It supplies the notion of structure-preserving transformation between vector spaces, and it is the concept whose generalization to several arguments — the multilinear map — leads directly to the definition of a tensor.

The Two Defining Properties

A function from one vector space to another is called a linear map, or linear transformation, if it satisfies two conditions for every pair of vectors and every scalar drawn from the underlying field: additivity, requiring that the map applied to the sum of two vectors equals the sum of the map applied to each vector individually, and homogeneity, requiring that the map applied to a scalar multiple of a vector equals the same scalar multiple of the map applied to that vector. These two conditions are frequently combined into a single requirement, that the map preserves arbitrary linear combinations.

The expression above states the combined additivity and homogeneity condition that defines linearity: a linear map applied to any linear combination of vectors equals the same linear combination of the map's values on those vectors.

Immediate Consequences

Several basic facts follow directly from the definition of a linear map. Every linear map sends the zero vector of its domain to the zero vector of its codomain, since the zero vector can be written as itself scaled by zero, and linearity forces the image to be scaled by zero as well. A linear map is completely determined by its action on a basis of its domain: because every vector in the domain can be written uniquely as a linear combination of basis vectors, and a linear map preserves linear combinations, knowing where the map sends each basis vector suffices to determine where it sends every vector in the space.

Matrix Representation

Once a basis is chosen for the domain and a basis is chosen for the codomain, a linear map between finite-dimensional vector spaces can be represented by a matrix, whose columns record the coordinates, relative to the codomain's basis, of the images of the domain's basis vectors. Applying the linear map to any vector then corresponds to multiplying the vector's coordinate representation by this matrix, converting the abstract operation of applying a linear map into ordinary matrix multiplication. Just as with vector coordinates, this matrix representation depends on the chosen bases, and changing either basis changes the matrix according to a specific transformation rule involving the relevant change-of-basis coefficients.

Special Types of Linear Maps

A linear map from a vector space to the scalar field itself, rather than to another vector space, is called a linear functional; the collection of all linear functionals on a given vector space forms its dual space, whose elements are covectors. A linear map from a vector space to itself is called a linear operator or endomorphism, and such maps are central to the study of eigenvalues, eigenvectors, and diagonalization. An invertible linear map, one that is both injective and surjective, is called a linear isomorphism, and two vector spaces connected by such a map are considered structurally identical for all purposes concerning their vector space operations.

From Linear Maps to Multilinear Maps and Tensors

The definition of a linear map generalizes directly to functions of several vector arguments: a multilinear map is required to be linear in each argument separately, holding the others fixed, and it is precisely this generalization that leads to the tensor product construction and, through it, to the formal definition of a tensor. In this sense, the linear map stands as the single-argument case from which the entire apparatus of multilinear algebra, and consequently tensor algebra, is built by successive generalization to more arguments.