In mathematics, a linear map (also called a linear mapping, transformation or, in some contexts, linear function) is a mapping V → W between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication.

An important special case is when V = W, an endomorphism of V, sometimes the term linear operator refers to this case^[1]. In another convention, linear operator allows V and W to differ, while requiring them to be real vector spaces^[2]. Sometimes the term linear function has the same meaning as linear map, while in analytic geometry it does not.

A linear map always maps linear subspaces onto linear subspaces (possibly of a lower dimension);^[3] for instance it maps a plane through the origin to a plane, straight line or point. Linear maps can often be represented as matrices, and simple examples include rotation and reflection linear transformations.

In the language of abstract algebra, a linear map is a module homomorphism. In the language of category theory it is a morphism in the category of modules over a given ring.

Thus, a linear map is said to be operation preserving. In other words, it does not matter whether the linear map is applied before or after the operations of addition and scalar multiplication.

If V and W are finite-dimensional vector spaces and a basis is defined for each vector space, then every linear map from V to W can be represented by a matrix.^[8] This is useful because it allows concrete calculations. Matrices yield examples of linear maps: if A is a real m × n matrix, then f (x) = Axdescribes a linear map** R→ R**m* (see Euclidean space).

Let {v1, …,** n*} be a basis for V. Then every vector* ** in V is uniquely determined by the coefficients c1, …, c**nin the field* R:

If f : V → W is a linear map,

which implies that the function f is entirely determined by the vectors f(v, …, f(vn). Now let {w1, …,** w**m} be a basis for* W*. Then we can represent each vector* f* (vj) as

Thus, the function f is entirely determined by the values of a**ij. If we put these values into an* m× nmatrix M*, then we can conveniently use it to compute the vector output of* ffor any vector in V*. To get* M*, every column* jof M* is a vector

corresponding to f( v s defined above. To define it more clearly, for some columnjthat corresponds to the mappingf (vj),

where M is the matrix of f. In other words, every column j = 1, …, n has a corresponding vector f (vj) whose coordinates* a1j*, …,* a**mjare the elements of column j*. A single linear map may be represented by many matrices. This is because the values of the elements of a matrix depend on the bases chosen.

The matrices of a linear transformation can be represented visually:

In two-dimensional space R 2 linear maps are described by 2 × 2 real matrices. These are some examples:

The composition of linear maps is linear: if f : V → W and g : W → Z are linear, then so is their composition g ∘ f : V → Z. It follows from this that the class of all vector spaces over a given field K, together with K-linear maps as morphisms, forms a category.

The inverse of a linear map, when defined, is again a linear map.

If f1 : V → W and f2 : V → W are linear, then so is their pointwise sum f1 + f2 (which is defined by (f1 + f2)(x) = f1(x) + f2(x)).

If f : V → W is linear and a is an element of the ground field K, then the map af, defined by (af)(x) = a (f (x)), is also linear.

Thus the set L (V, W) of linear maps from V to W itself forms a vector space over K, sometimes denoted Hom(V, W). Furthermore, in the case that V = W, this vector space (denoted End(V)) is an associative algebra under composition of maps, since the composition of two linear maps is again a linear map, and the composition of maps is always associative. This case is discussed in more detail below.

Given again the finite-dimensional case, if bases have been chosen, then the composition of linear maps corresponds to the matrix multiplication, the addition of linear maps corresponds to the matrix addition, and the multiplication of linear maps with scalars corresponds to the multiplication of matrices with scalars.

A linear transformation f: V → V is an endomorphism of V; the set of all such endomorphisms End(V) together with addition, composition and scalar multiplication as defined above forms an associative algebra with identity element over the field K (and in particular a ring). The multiplicative identity element of this algebra is the identity map id: V → V.

An endomorphism of V that is also an isomorphism is called an automorphism of V. The composition of two automorphisms is again an automorphism, and the set of all automorphisms of V forms a group, the automorphism group of V which is denoted by Aut(V) or GL(V). Since the automorphisms are precisely those endomorphisms which possess inverses under composition, Aut(V) is the group of units in the ring End(V).

If V has finite dimension n, then End(V) is isomorphic to the associative algebra of all n × n matrices with entries in K. The automorphism group of V is isomorphic to the general linear group GL(n, K) of all n × n invertible matrices with entries in K.

If f : V → W is linear, we define the kernel and the image or range of f by

ker(f) is a subspace of V and im(f) is a subspace of W. The following dimension formula is known as the rank–nullity theorem:

The number dim(im(f)) is also called the rank of f and written as rank(f), or sometimes, ρ(f); the number dim(ker(f)) is called the nullity of f and written as null(f) or ν(f). If V and W are finite-dimensional, bases have been chosen and f is represented by the matrix A, then the rank and nullity of f are equal to the rank and nullity of the matrix A, respectively.

This is the dual notion to the kernel: just as the kernel is a subspace of the domain, the co-kernel is a quotient space*space]]the Formally, one has the exact sequence

These can be interpreted thus: given a linear equation f (v) = w to solve,

The dimension of the co-kernel and the dimension of the image (the rank) add up to the dimension of the target space.

For finite dimensions, this means that the dimension of the quotient space W/f (V) is the dimension of the target space minus the dimension of the image.

For a linear operator with finite-dimensional kernel and co-kernel, one may define index as:

namely the degrees of freedom minus the number of constraints.

For a transformation between finite-dimensional vector spaces, this is just the difference dim(V) − dim(W), by rank–nullity. This gives an indication of how many solutions or how many constraints one has: if mapping from a larger space to a smaller one, the map may be onto, and thus will have degrees of freedom even without constraints. Conversely, if mapping from a smaller space to a larger one, the map cannot be onto, and thus one will have constraints even without degrees of freedom.

The index of an operator is precisely the Euler characteristic of the 2-term complex 0 → V → W → 0. In operator theory, the index of Fredholm operators is an object of study, with a major result being the Atiyah–Singer index theorem.^[10]

No classification of linear maps could hope to be exhaustive.

The following incomplete list enumerates some important classifications that do not require any additional structure on the vector space.

Let V and W denote vector spaces over a field, F. Let T: V → W be a linear map.

Given a linear map which is an endomorphism whose matrix is A, in the basis B of the space it transforms vector coordinates [u] as [v] = A [u]. As vectors change with the inverse of B (vectors are contravariant) its inverse transformation is [v] = B [v'].

Substituting this in the first expression

hence

Therefore, the matrix in the new basis is A′ = B−1AB, being B the matrix of the given basis.

Therefore, linear maps are said to be 1-co- 1-contra-variant objects, or type (1, 1) tensors.

A linear transformation between topological vector spaces, for example normed spaces, may be continuous. If its domain and codomain are the same, it will then be a continuous linear operator. A linear operator on a normed linear space is continuous if and only if it is bounded, for example, when the domain is finite-dimensional.^[11] An infinite-dimensional domain may have discontinuous linear operators.

An example of an unbounded, hence discontinuous, linear transformation is differentiation on the space of smooth functions equipped with the supremum norm (a function with small values can have a derivative with large values, while the derivative of 0 is 0).

For a specific example, sin(nx)/nconverges to 0, but its derivative cos(nx) does not, so differentiation is not continuous at 0 (and by a variation of this argument, it is not continuous anywhere).

A specific application of linear maps is for geometric transformations, such as those performed in computer graphics, where the translation, rotation and scaling of 2D or 3D objects is performed by the use of a transformation matrix. Linear mappings also are used as a mechanism for describing change: for example in calculus correspond to derivatives; or in relativity, used as a device to keep track of the local transformations of reference frames.

Another application of these transformations is in compiler optimizations of nested-loop code, and in parallelizing compiler techniques.