In mathematics, a linear map (also called a linear transformation, linear function or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. The expression "linear operator" is commonly used for linear maps from a vector space to itself (endomorphisms). In advanced mathematics, the definition of linear function coincides with the definition of linear map.
In the language of abstract algebra, a linear map is a homomorphism of vector spaces, or in the language of category theory a morphism in KVect, the category of vector spaces over a given field K.
Let V and W be vector spaces over the same field K. A function f : V → W is said to be a linear map if for any two vectors x and y in V and any scalar a in K, the following two conditions are satisfied:
additivity  
homogeneity of degree 1. 
This is equivalent to requiring that for any vectors x_{1}, ..., x_{m} and scalars a_{1}, ..., a_{m}, the equality
holds.
It immediately follows from the definition that f(0) = 0.
Occasionally, V and W can be considered to be vector spaces over different fields. It is then necessary to specify which of these ground fields is being used in the definition of "linear". If V and W are considered as spaces over the field K as above, we talk about Klinear maps. For example, the conjugation of complex numbers is an Rlinear map C → C, but it is not Clinear.
A linear map from V to K (with K viewed as a vector space over itself) is called a linear functional.
If V and W are finitedimensional, and one has chosen bases in those spaces, then every linear map from V to W can be represented as a matrix; this is useful because it allows concrete calculations. Conversely, matrices yield examples of linear maps: if A is a real mbyn matrix, then the rule f(x) = Ax describes a linear map R^{n} → R^{m} (see Euclidean space).
Let be a basis for V. Then every vector v in V is uniquely determined by the coefficients in
If f : V → W is a linear map,
which implies that the function f is entirely determined by the values of
Now let be a basis for W. Then we can represent the values of each f(v_{j}) as
Thus, the function f is entirely determined by the values of a_{i,j}.
If we put these values into an mbyn matrix M, then we can conveniently use it to compute the value of f for any vector in V. For if we place the values of in an nby1 matrix C, we have MC = the mby1 matrix whose i.th element is the coordinate of f(v) which belongs to the base w_{i}.
A single linear map may be represented by many matrices. This is because the values of the elements of the matrix depend on the bases that are chosen.
Some special cases of linear transformations of twodimensional space R^{2} are illuminating:
The composition of linear maps is linear: if f : V → W and g : W → Z are linear, then so is their composition g o f : V → Z. It follows from this that the class of all vector spaces over a given field K, together with Klinear maps as morphisms, forms a category.
The inverse of a linear map, when defined, is again a linear map.
If f_{1} : V → W and f_{2} : V → W are linear, then so is their sum f_{1} + f_{2} (which is defined by (f_{1} + f_{2})(x) = f_{1}(x) + f_{2}(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 finitedimensional 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 by n matrices with entries in K. The automorphism group of V is isomorphic to the general linear group GL(n, K) of all n by 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, known as the ranknullity theorem, is often useful:
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 finitedimensional, 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.
A subtler invariant of a linear transformation is the cokernel, which is defined as
This is the dual notion to the kernel: just as the kernel is a subspace of the domain, the cokernel is a quotient space of the target. Formally, one has the exact sequence
These can be interpreted thus: given a linear equation f(v) = w to solve,
The dimension of the cokernel 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.
As a simple example, consider the map given by f(x,y) = (0,y). Then for an equation f(x,y) = (a,b) to have a solution, we must have a = 0 (one constraint), and in that case the solution space is (x,b), or equivalently stated, (0,b) + (x,0), (one degree of freedom). The kernel may be expressed as the subspace (x,0) < V: the value of x is the freedom in a solution – while the cokernel may be expressed via the map given a vector (a,b), the value of a is the obstruction to there being a solution.
An example illustrating the infinitedimensional case is afforded by the map with b_{1} = 0 and b_{n + 1} = a_{n} for n > 0. Its image consists of all sequences with first element 0, and thus its cokernel consists of the classes of sequences with identical first element. Thus, whereas its kernel has dimension 0 (it maps only the zero sequence to the zero sequence), its cokernel has dimension 1. Since the domain and the target space are the same, the rank and the dimension of the kernel add up to the same sum as the rank and the dimension of the cokernel ( ), but in the infinitedimensional case it cannot be inferred that the kernel and the cokernel of an endomorphism have the same dimension (). The reverse situation obtains for the map with c_{n} = a_{n + 1}. Its image is the entire target space, and hence its cokernel has dimension 0, but since it maps all sequences in which only the first element is nonzero to the zero sequence, its kernel has dimension 1.
For a linear operator with finitedimensional kernel and cokernel, one may define index as:
namely the degrees of freedom minus the number of constraints.
For a transformation between finitedimensional vector spaces, this is just the difference dimV − dimW, by ranknullity. 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 comes of its own in infinite dimensions: it is how homology is defined, which is a central theory in algebra and algebraic topology; the index of an operator is precisely the Euler characteristic of the 2term complex In operator theory, the index of Fredholm operators is an object of study, with a major result being the Atiyah–Singer index theorem.
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.
A linear operator between topological vector spaces, for example normed spaces, may also be continuous and therefore be a continuous linear operator. On a normed space, a linear operator is continuous if and only if it is bounded, for example, when the domain is finitedimensional. If the domain is infinitedimensional, then there may be discontinuous linear operators. An example of an unbounded, hence not continuous, 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).
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.
Another application of these transformations is in compiler optimizations of nestedloop code, and in parallelizing compiler techniques.

