В математике , А линейное отображение (также называется линейное отображение , линейное преобразование , векторное пространство , гомоморфизм , или в некоторых контекстах линейных функций ) представляет собой отображение между двумя векторными пространствами , сохраняющими операции сложения векторов и скалярного умножения . Те же имена и определение используются и для более общего случая модулей над кольцом ; см. Гомоморфизм модулей .
Если линейное отображение является биекцией, то это называется линейным изоморфизмом . В случае, когда, линейное отображение называется (линейным) эндоморфизмом . Иногда термин « линейный оператор» относится к этому случаю [1], но термин «линейный оператор» может иметь разные значения для разных соглашений: например, его можно использовать, чтобы подчеркнуть, что а также являются вещественными векторными пространствами (не обязательно с), [ необходима цитата ] или его можно использовать, чтобы подчеркнуть, что- это функциональное пространство , которое является общепринятым в функциональном анализе . [2] Иногда термин линейная функция имеет то же значение, что и линейная карта , хотя в анализе это не так.
Линейное отображение из V в W всегда отображает происхождение V к происхождению W . Более того, он отображает линейные подпространства в V на линейные подпространства в W (возможно, меньшей размерности ); [3] , например, он отображает плоскость через происхождения в V в любой плоскости , проходящей через начало координат в W , в линии через начало координат в W , или просто координат в W . Линейные карты часто могут быть представлены в виде матриц , а простые примеры включают линейные преобразования вращения и отражения .
На языке теории категорий линейные отображения - это морфизмы векторных пространств.
Определение и первые следствия
Позволять а также быть векторными пространствами над одним и тем же полем . Функцияназывается линейным отображением, если для любых двух векторов и любой скаляр выполняются следующие два условия:
аддитивность / операция сложения | |
однородность степени 1 / операция скалярного умножения |
Таким образом, линейное отображение называется сохраняющим операцию . Другими словами, не имеет значения, применяется ли линейная карта до (правые части приведенных выше примеров) или после (левые части примеров) операций сложения и скалярного умножения.
По ассоциативности операции сложения, обозначенной как +, для любых векторов и скаляры выполняется следующее равенство: [4] [5]
Обозначая нулевые элементы векторных пространств а также от а также соответственно, отсюда следует, что Позволять а также в уравнении однородности степени 1:
Изредка, а также могут быть векторными пространствами над разными полями. Затем необходимо указать, какое из этих основных полей используется в определении «линейного». Если а также пробелы над одним и тем же полем как указано выше, тогда мы говорим о -линейные карты. Так , например, конъюгации из комплексных чисел является-линейная карта , но это не так -линейный, где а также являются символами, представляющими наборы действительных чисел и комплексных чисел соответственно.
Линейная карта с участием рассматриваемое как одномерное векторное пространство над самим собой, называется линейным функционалом . [6]
Эти утверждения обобщаются на любой левый модуль над кольцом без модификации, и любому правому модулю при обращении скалярного умножения.
Примеры
- Типичным примером, который дает линейным картам их имя, является функция , график которой представляет собой линию, проходящую через начало координат. [7]
- В общем, любая гомотетия где с центром в начале векторного пространства является линейная карта.
- Нулевая карта между двумя векторными пространствами (над одним и тем же полем ) является линейным.
- Тождественное отображение на любом модуле является линейным оператором.
- Для вещественных чисел карта не является линейным.
- Для вещественных чисел карта не является линейным (а является аффинным преобразованием ).
- Если это вещественная матрица , тогда определяет линейную карту из к отправив вектор-столбец к вектору-столбцу . И наоборот, любое линейное отображение между конечномерными векторными пространствами может быть представлено таким образом; см. § Матрицы ниже.
- Если является изометрией между вещественными нормированными пространствами такими, что тогда является линейным отображением. Этот результат не обязательно верен для сложного нормированного пространства. [8]
- Дифференцирование определяет линейное отображение пространства всех дифференцируемых функций в пространство всех функций. Он также определяет линейный оператор в пространстве всех гладких функций (линейный оператор - это линейный эндоморфизм , то есть линейное отображение, область определения и область значений которого совпадают). Примером является
- Определенный интеграл по некоторому интервалу I - это линейное отображение пространства всех действительных интегрируемых функций на I в. Например,
- 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 § Matrices (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 a 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 function with is a linear map. This function scales the component of a vector by the factor .
The function is additive: It doesn't matter whether vectors are first added and then mapped or whether they are mapped and finally added:
The function is homogeneous: It doesn't matter whether a vector is first scaled and then mapped or first mapped and then scaled:
Матрицы
If and are finite-dimensional vector spaces and a basis is defined for each vector space, then every linear map from to can be represented by a matrix.[9] This is useful because it allows concrete calculations. Matrices yield examples of linear maps: if is a real matrix, then describes a linear map (see Euclidean space).
Let be a basis for . Then every vector is uniquely determined by the coefficients in the field :
If is a linear map,
which implies that the function f is entirely determined by the vectors . Now let be a basis for . Then we can represent each vector as
Thus, the function is entirely determined by the values of . If we put these values into an matrix , then we can conveniently use it to compute the vector output of for any vector in . To get , every column of is a vector
corresponding to as defined above. To define it more clearly, for some column that corresponds to the mapping ,
where is the matrix of . In other words, every column has a corresponding vector whose coordinates are the elements of column . 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:
- Matrix for relative to :
- Matrix for relative to :
- Transition matrix from to :
- Transition matrix from to :
Such that starting in the bottom left corner and looking for the bottom right corner , one would left-multiply—that is, . The equivalent method would be the "longer" method going clockwise from the same point such that is left-multiplied with , or .
Examples in dimension two
In two-dimensional space R2 linear maps are described by 2 × 2 matrices. These are some examples:
- rotation
- by 90 degrees counterclockwise:
- by an angle θ counterclockwise:
- by 90 degrees counterclockwise:
- reflection
- through the x axis:
- through the y axis:
- through a line making an angle θ with the origin:
- through the x axis:
- scaling by 2 in all directions:
- horizontal shear mapping:
- squeeze mapping:
- projection onto the y axis:
Векторное пространство линейных отображений
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 ,[10] sometimes denoted .[11] 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.
Endomorphisms and automorphisms
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 .
Ядро, образ и теорема ранга – недействительности
If is linear, we define the kernel and the image or range of by
is a subspace of and is a subspace of . The following dimension formula is known as the rank–nullity theorem:
- [12]
The number is also called the rank of and written as , or sometimes, ;[13][14] the number is called the nullity of and written as or .[13][14] If and are finite-dimensional, bases have been chosen and is represented by the matrix , then the rank and nullity of are equal to the rank and nullity of the matrix , respectively.
Cokernel
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 co-kernel 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 kernel is the space of solutions to the homogeneous equation f(v) = 0, and its dimension is the number of degrees of freedom in the space of solutions, if it is not empty;
- the co-kernel is the space of constraints that the solutions must satisfy, and its dimension is the maximal number of independent constraints.
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.
As a simple example, consider the map f: R2 → R2, 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 W → R, given a vector (a, b), the value of a is the obstruction to there being a solution.
An example illustrating the infinite-dimensional case is afforded by the map f: R∞ → R∞, with b1 = 0 and bn + 1 = an 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 co-kernel 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 co-kernel ( ), but in the infinite-dimensional case it cannot be inferred that the kernel and the co-kernel of an endomorphism have the same dimension (0 ≠ 1). The reverse situation obtains for the map h: R∞ → R∞, with cn = an + 1. Its image is the entire target space, and hence its co-kernel has dimension 0, but since it maps all sequences in which only the first element is non-zero to the zero sequence, its kernel has dimension 1.
Index
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.[15]
Алгебраические классификации линейных преобразований
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.
Definition: T is said to be injective or a monomorphism if any of the following equivalent conditions are true:
- T is one-to-one as a map of sets.
- ker T = {0V}
- dim(ker T) = 0
- T is monic or left-cancellable, which is to say, for any vector space U and any pair of linear maps R: U → V and S: U → V, the equation TR = TS implies R = S.
- T is left-invertible, which is to say there exists a linear map S: W → V such that ST is the identity map on V.
Definition: T is said to be surjective or an epimorphism if any of the following equivalent conditions are true:
- T is onto as a map of sets.
- coker T = {0W}
- T is epic or right-cancellable, which is to say, for any vector space U and any pair of linear maps R: W → U and S: W → U, the equation RT = ST implies R = S.
- T is right-invertible, which is to say there exists a linear map S: W → V such that TS is the identity map on W.
Definition: T is said to be an isomorphism if it is both left- and right-invertible. This is equivalent to T being both one-to-one and onto (a bijection of sets) or also to T being both epic and monic, and so being a bimorphism.
If T: V → V is an endomorphism, then:
- If, for some positive integer n, the n-th iterate of T, Tn, is identically zero, then T is said to be nilpotent.
- If T2 = T, then T is said to be idempotent
- If T = kI, where k is some scalar, then T is said to be a scaling transformation or scalar multiplication map; see scalar matrix.
Смена основы
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.
Непрерывность
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.[16] 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).
Приложения
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.
Смотрите также
- Antilinear map
- Bent function
- Bounded operator
- Continuous linear operator
- Linear functional
- Linear isometry
Заметки
- ^ "Linear transformations of V into V are often called linear operators on V." Rudin 1976, p. 207
- ^ Let V and W be two real vector spaces. A mapping a from V into W Is called a 'linear mapping' or 'linear transformation' or 'linear operator' [...] from V into W, if
for all ,
for all and all real λ. Bronshtein & Semendyayev 2004, p. 316 - ^ Rudin 1991, p. 14
Here are some properties of linear mappings whose proofs are so easy that we omit them; it is assumed that and :- If A is a subspace (or a convex set, or a balanced set) the same is true of
- If B is a subspace (or a convex set, or a balanced set) the same is true of
- In particular, the set:
- ^ Rudin 1991, p. 14. Suppose now that X and Y are vector spaces over the same scalar field. A mapping is said to be linear if for all and all scalars and . Note that one often writes , rather than , when is linear.
- ^ Rudin 1976, p. 206. A mapping A of a vector space X into a vector space Y is said to be a linear transformation if: for all and all scalars c. Note that one often writes instead of if A is linear.
- ^ Rudin 1991, p. 14. Linear mappings of X onto its scalar field are called linear functionals.
- ^ "terminology - What does 'linear' mean in Linear Algebra?". Mathematics Stack Exchange. Retrieved 2021-02-17.
- ^ Wilansky 2013, pp. 21-26.
- ^ Rudin 1976, p. 210 Suppose and are bases of vector spaces X and Y, respectively. Then every determines a set of numbers such that
- ^ Axler (2015) p. 52, § 3.3
- ^ Tu (2011), p. 19, § 3.1
- ^ Horn & Johnson 2013, 0.2.3 Vector spaces associated with a matrix or linear transformation, p. 6
- ^ a b Katznelson & Katznelson (2008) p. 52, § 2.5.1
- ^ a b Halmos (1974) p. 90, § 50
- ^ Nistor, Victor (2001) [1994], "Index theory", Encyclopedia of Mathematics, EMS Press: "The main question in index theory is to provide index formulas for classes of Fredholm operators ... Index theory has become a subject on its own only after M. F. Atiyah and I. Singer published their index theorems"
- ^ Rudin 1991, p. 15 1.18 Theorem Let be a linear functional on a topological vector space X. Assume for some . Then each of the following four properties implies the other three:
- is continuous
- The null space is closed.
- is not dense in X.
- is bounded in some neighbourhood V of 0.
Библиография
- Axler, Sheldon Jay (2015). Linear Algebra Done Right (3rd ed.). Springer. ISBN 978-3-319-11079-0.
- Bronshtein, I. N.; Semendyayev, K. A. (2004). Handbook of Mathematics (4th ed.). New York: Springer-Verlag. ISBN 3-540-43491-7.
- Halmos, Paul Richard (1974) [1958]. Finite-Dimensional Vector Spaces (2nd ed.). Springer. ISBN 0-387-90093-4.
- Horn, Roger A.; Johnson, Charles R. (2013). Matrix Analysis (Second ed.). Cambridge University Press. ISBN 978-0-521-83940-2.
- Katznelson, Yitzhak; Katznelson, Yonatan R. (2008). A (Terse) Introduction to Linear Algebra. American Mathematical Society. ISBN 978-0-8218-4419-9.
- Lang, Serge (1987), Linear Algebra (Third ed.), New York: Springer-Verlag, ISBN 0-387-96412-6
- Rudin, Walter (1973). Functional Analysis. International Series in Pure and Applied Mathematics. 25 (First ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 9780070542259.
- Rudin, Walter (1976). Principles of Mathematical Analysis. Walter Rudin Student Series in Advanced Mathematics (3rd ed.). New York: McGraw–Hill. ISBN 978-0-07-054235-8.
- Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
- Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
- Swartz, Charles (1992). An introduction to Functional Analysis. New York: M. Dekker. ISBN 978-0-8247-8643-4. OCLC 24909067.
- Tu, Loring W. (2011). An Introduction to Manifolds (2nd ed.). Springer. ISBN 978-0-8218-4419-9.
- Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114.