Сызықтық функциялар және векторлық кеңістіктер

Mathematical function, in linear algebra

In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication. The same names and the same definition are also used for the more general case of modules over a ring; see Module homomorphism. If a linear map is a bijection then it is called a Linear isomorphism. In the case where , a linear map is called a linear endomorphism. Sometimes the term Linear operator refers to this case, but the term "linear operator" can have different meanings for different conventions: for example, it can be used to emphasize that and are real vector spaces (not necessarily with ), or it can be used to emphasize that is a function space, which is a common convention in functional analysis. Sometimes the term linear function has the same meaning as linear map, while in analysis it does not. A linear map from to always maps the origin of to the origin of Moreover, it maps linear subspaces in onto linear subspaces in (possibly of a lower dimension); for example, it maps a plane through the origin in to either a plane through the origin in , a line through the origin in , or just the origin in Linear maps can often be represented as matrices, and simple examples include rotation and reflection linear transformations. In the language of category theory, linear maps are the morphisms of vector spaces.

A prototypical example that gives linear maps their name is a function , of which the graph is a line through the origin. More generally, any homothety centered in the origin of a vector space is a linear map (here c is a scalar). The zero map between two vector spaces (over the same field) is linear. The identity map on any module is a linear operator. For real numbers, the map is not linear. For real numbers, the map is not linear (but is an affine transformation). If is a real matrix, then defines a linear map from to by sending a column vector to the column vector Conversely, any linear map between finite dimensional vector spaces can be represented in this manner; see the , below. If is an isometry between real normed spaces such that then is a linear map. This result is not necessarily true for complex normed space. Differentiation defines a linear map from the space of all differentiable functions to the space of all functions. It also defines a linear operator on the space of all smooth functions (a linear operator is a linear endomorphism, that is, a linear map with the same domain and codomain). Indeed,
A definite integral over some interval I is a linear map from the space of all real valued integrable functions on I to Indeed,
An indefinite integral (or antiderivative) with a fixed integration starting point defines a linear map from the space of all real valued integrable functions on to the space of all real valued, differentiable functions on Without a fixed starting point, the antiderivative maps to the quotient space of the differentiable functions by the linear space of constant functions. If and are finite dimensional vector spaces over a field F, of respective dimensions m and n, then the function that maps linear maps to n × m matrices in the way described in (below) is a linear map, and even a linear isomorphism. The expected value of a random variable (which is in fact a function, and as such an element of a vector space) is linear, as for random variables and we have and , but the variance of a random variable is not linear.

The composition of linear maps is linear: if and are linear, then so is their composition 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 and are linear, then so is their pointwise sum , which is defined by
If is linear and is an element of the ground field , then the map , defined by , is also linear. Thus the set of linear maps from to itself forms a vector space over , sometimes denoted Furthermore, in the case that , this vector space, denoted , 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 is an endomorphism of ; the set of all such endomorphisms together with addition, composition and scalar multiplication as defined above forms an associative algebra with identity element over the field (and in particular a ring). The multiplicative identity element of this algebra is the identity map
An endomorphism of that is also an isomorphism is called an automorphism of The composition of two automorphisms is again an automorphism, and the set of all automorphisms of forms a group, the automorphism group of which is denoted by or Since the automorphisms are precisely those endomorphisms which possess inverses under composition, is the group of units in the ring
If has finite dimension , then is isomorphic to the associative algebra of all matrices with entries in The automorphism group of is isomorphic to the general linear group of all invertible matrices with entries in .

No classification of linear maps could 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 and let T: V → W be a linear map.

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. 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)/n converges 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.