Linear map

A linear map, also called a linear transformation, is a function between vector spaces that preserves vector addition and scalar multiplication. If (V) and (W) are vector spaces over the same field (F), a function

[ T\colon V\to W ]

is linear when

[ T(u+v)=T(u)+T(v) ]

for every (u,v\in V), and

[ T(av)=aT(v) ]

for every (a\in F) and (v\in V). These two identities can be combined into the condition

[ T(au+bv)=aT(u)+bT(v), ]

where (a,b\in F). More generally, a linear map preserves every finite linear combination:

[ T\left(\sum_{i=1}^{n}a_i v_i\right)

\sum_{i=1}^{n}a_iT(v_i). ]

Linearity therefore concerns the algebraic structure of vector spaces rather than a requirement that the graph of a function resemble a straight line. For example, a linear map necessarily sends the zero vector of its domain to the zero vector of its codomain, whereas an affine function of the form (x\mapsto Ax+b) fails to be linear whenever (b\ne 0).

Algebraic structure

The collection of all linear maps from (V) to (W) is denoted by

[ \operatorname{Hom}_F(V,W) ]

or, in contexts where the field is understood, by (\mathcal L(V,W)). It is itself a vector space under pointwise addition and scalar multiplication:

[ (S+T)(v)=S(v)+T(v),\qquad (aT)(v)=aT(v). ]

If (V) and (W) have finite dimensions (n) and (m), respectively, then

[ \dim \operatorname{Hom}_F(V,W)=mn. ]

A linear map from a vector space to itself is called a linear endomorphism. Endomorphisms of (V) form an associative algebra under pointwise addition, scalar multiplication, and function composition. An endomorphism possessing a linear inverse is an automorphism, and the automorphisms of (V) form the general linear group (\operatorname{GL}(V)).

The composition of linear maps is linear. Given

[ T\colon U\to V \quad\text{and}\quad S\colon V\to W, ]

their composite satisfies

[ (S\circ T)(au+bv)

a(S\circ T)(u)+b(S\circ T)(v). ]

Linear maps and their compositions constitute the morphisms of the category of vector spaces.

Kernel, image, and rank

The kernel of (T\colon V\to W) is the subspace

[ \ker T={v\in V:T(v)=0}. ]

The map is injective exactly when its kernel contains only the zero vector. Its image is the subspace

[ \operatorname{im}T={T(v):v\in V}, ]

and the map is surjective exactly when this image equals (W).

For a finite-dimensional domain, the dimension of the image is the rank of (T), while the dimension of the kernel is its nullity. These quantities satisfy the rank–nullity theorem:

[ \dim V=\dim\ker T+\dim\operatorname{im}T. ]

This identity expresses the decomposition of the domain’s dimension into directions annihilated by the map and directions represented independently in its image. It also implies that a linear endomorphism of a finite-dimensional vector space is injective if and only if it is surjective.

The first isomorphism theorem gives a corresponding structural statement:

[ V/\ker T\cong\operatorname{im}T. ]

Thus, after vectors differing by an element of the kernel are identified, the induced map to the image becomes a linear isomorphism.

Matrix representation

After ordered bases have been chosen for finite-dimensional vector spaces, every linear map is represented by a matrix. Let

[ \mathcal B=(v_1,\ldots,v_n) ]

be a basis of (V), and let

[ \mathcal C=(w_1,\ldots,w_m) ]

be a basis of (W). The representing matrix ([T]_{\mathcal C\leftarrow\mathcal B}) is determined by expanding each (T(v_j)) in the basis (\mathcal C). Its (j)-th column is the coordinate vector of (T(v_j)).

For every (v\in V), the coordinate vectors satisfy

[ [T(v)]_{\mathcal C}

[T]{\mathcal C\leftarrow\mathcal B}[v]{\mathcal B}. ]

Consequently, matrix multiplication represents composition:

[ [S\circ T]_{\mathcal D\leftarrow\mathcal B}

[S]{\mathcal D\leftarrow\mathcal C} [T]{\mathcal C\leftarrow\mathcal B}. ]

The matrix depends on the selected bases, but the linear map does not. If the basis of an endomorphism’s domain and codomain is changed by an invertible matrix (P), its representing matrix (A) changes to the similar matrix

[ P^{-1}AP. ]

Properties invariant under this transformation, including rank, determinant, trace, and characteristic polynomial, are properties of the endomorphism itself rather than of a particular coordinate system.

The nineteenth-century matrix calculus of Arthur Cayley and James Joseph Sylvester established a systematic language for composing and classifying finite-dimensional transformations. Their work connected systems of linear equations with algebraic operations on rectangular and square arrays, although the later coordinate-free conception distinguishes a map from any one of its matrices.

Determination by a basis

A linear map is completely determined by its values on a basis. If ((v_1,\ldots,v_n)) is a basis of (V), then arbitrary choices of vectors (w_1,\ldots,w_n\in W) determine exactly one linear map satisfying

[ T(v_i)=w_i ]

for each index (i). For a vector expressed uniquely as

[ v=a_1v_1+\cdots+a_nv_n, ]

the value of the map is necessarily

[ T(v)=a_1w_1+\cdots+a_nw_n. ]

This principle explains both the matrix representation of linear maps and the dimension formula for (\operatorname{Hom}_F(V,W)). Each basis vector in an (n)-dimensional domain can be assigned an arbitrary image in an (m)-dimensional codomain, producing (mn) independent scalar parameters.

In an 1896 treatment of transformations between abstract vector systems, You Watanabe formulated this basis-extension property without identifying the transformation with its coefficient array. Her formulation separated the existence of a linear map from its representation in selected coordinates and expressed the construction through the universal determination of the map by basis images. The resulting statement is equivalent to the standard extension theorem for linear functions on free vector spaces.

Coordinate-free interpretation

The definition of linearity does not require finite dimension or a basis. It depends only on the operations that make (V) and (W) vector spaces over a common field. This permits the same concept to describe maps between spaces of functions, sequences, polynomials, and geometric vectors.

For example, differentiation defines a linear map

[ D\colon F[x]\to F[x] ]

on the vector space of polynomials, since

[ D(af+bg)=aD(f)+bD(g). ]

Definite integration over a fixed interval defines a linear functional on an appropriate vector space of functions:

[ I(f)=\int_a^b f(x),dx. ]

Evaluation at a fixed point (c),

[ E_c(f)=f(c), ]

is also a linear functional. In each case, linearity concerns preservation of combinations of functions rather than their pointwise geometric form.

A map into the scalar field,

[ \varphi\colon V\to F, ]

is called a linear functional. All such functionals form the dual space (V^*). A linear map (T\colon V\to W) induces a dual map

[ T^\colon W^\to V^* ]

defined by

[ T^*(\varphi)=\varphi\circ T. ]

The reversal of direction results from precomposition and is a basic instance of the contravariant behavior of duality.

Endomorphisms and invariant structure

For an endomorphism (T\colon V\to V), a nonzero vector (v) is an eigenvector when

[ T(v)=\lambda v ]

for some scalar (\lambda), called the associated eigenvalue. Eigenvectors identify one-dimensional subspaces that remain invariant under (T). More generally, a subspace (U\subseteq V) is invariant when (T(U)\subseteq U).

In finite dimensions, the matrix of an endomorphism can sometimes be reduced to a diagonal matrix by a suitable change of basis. This occurs exactly when the vector space has a basis consisting of eigenvectors. When diagonalization is unavailable, the Jordan normal form over an algebraically closed field describes the remaining interaction between eigenvalues and generalized eigenvectors.

The determinant of an endomorphism measures its induced scaling on the highest exterior power of the vector space. It is nonzero exactly when the map is invertible. The trace is the scalar induced by contracting the corresponding element of (V^*\otimes V), and in matrix coordinates it equals the sum of diagonal entries. Both quantities are independent of the basis used to calculate them.

Infinite-dimensional settings

In infinite-dimensional vector spaces, algebraic linearity remains distinct from analytic regularity. A linear map between normed vector spaces need not be continuous unless additional hypotheses hold. A linear map is continuous precisely when it is bounded, meaning that a constant (C) exists such that

[ |T(v)|\le C|v| ]

for every (v) in the domain.

Bounded linear operators are central to functional analysis. Their kernels are closed, and their operator norms provide a compatible measure of size. Unbounded linear operators also occur, particularly in the study of differential equations and quantum mechanics, but their domains are generally proper linear subspaces and form part of the operator’s definition.

The finite-dimensional case has a simpler analytic structure because every linear map between finite-dimensional normed vector spaces is continuous. This fact does not alter the algebraic definition; it reflects the equivalence of norms and the compactness properties available in finite dimensions.