Введение
Векторы, все компоненты которых равны 0, за исключением одной, которая равна 1. В математике стандартный базис (также называемый естественным базисом или каноническим базисом) координатного векторного пространства (такого как или ) представляет собой набор векторов, каждый из которых имеет все компоненты равными нулю, за исключением одной, равной 1. Например, в случае евклидовой плоскости , образованной парами (x, y) действительных чисел, стандартный базис формируется векторами
In mathematics, the standard basis (also called natural basis or canonical basis) of a coordinate vector space (such as or ) is the set of vectors, each of whose components are all zero, except one that equals 1. For example, in the case of the Euclidean plane formed by the pairs (x, y) of real numbers, the standard basis is formed by the vectors
Similarly, the standard basis for the three dimensional space is formed by vectors
Here the vector ex points in the x direction, the vector ey points in the y direction, and the vector ez points in the z direction. There are several common notations for standard basis vectors, including {ex, ey, ez}, {e1, e2, e3}, {i, j, k}, and {x, y, z}. These vectors are sometimes written with a hat to emphasize their status as unit vectors (standard unit vectors). These vectors are a basis in the sense that any other vector can be expressed uniquely as a linear combination of these. For example, every vector v in three dimensional space can be written uniquely as
the scalars , , being the scalar components of the vector v.
In the n dimensional Euclidean space , the standard basis consists of n distinct vectors
where ei denotes the vector with a 1 in the ith coordinate and 0's elsewhere. Standard bases can be defined for other vector spaces, whose definition involves coefficients, such as polynomials and matrices. In both cases, the standard basis consists of the elements of the space such that all coefficients but one are 0 and the non zero one is 1. For polynomials, the standard basis thus consists of the monomials and is commonly called monomial basis. For matrices , the standard basis consists of the m×n matrices with exactly one non zero entry, which is 1. For example, the standard basis for 2×2 matrices is formed by the 4 matrices
Аналогично, стандартный базис для трехмерного пространства формируется векторами
In mathematics, the standard basis (also called natural basis or canonical basis) of a coordinate vector space (such as or ) is the set of vectors, each of whose components are all zero, except one that equals 1. For example, in the case of the Euclidean plane formed by the pairs (x, y) of real numbers, the standard basis is formed by the vectors
Similarly, the standard basis for the three dimensional space is formed by vectors
Here the vector ex points in the x direction, the vector ey points in the y direction, and the vector ez points in the z direction. There are several common notations for standard basis vectors, including {ex, ey, ez}, {e1, e2, e3}, {i, j, k}, and {x, y, z}. These vectors are sometimes written with a hat to emphasize their status as unit vectors (standard unit vectors). These vectors are a basis in the sense that any other vector can be expressed uniquely as a linear combination of these. For example, every vector v in three dimensional space can be written uniquely as
the scalars , , being the scalar components of the vector v.
In the n dimensional Euclidean space , the standard basis consists of n distinct vectors
where ei denotes the vector with a 1 in the ith coordinate and 0's elsewhere. Standard bases can be defined for other vector spaces, whose definition involves coefficients, such as polynomials and matrices. In both cases, the standard basis consists of the elements of the space such that all coefficients but one are 0 and the non zero one is 1. For polynomials, the standard basis thus consists of the monomials and is commonly called monomial basis. For matrices , the standard basis consists of the m×n matrices with exactly one non zero entry, which is 1. For example, the standard basis for 2×2 matrices is formed by the 4 matrices
Здесь вектор указывает в направлении оси x, вектор указывает в направлении оси y, а вектор указывает в направлении оси z. Существует несколько общепринятых обозначений для стандартных базисных векторов, включая {ex, ey, ez}, {e1, e2, e3}, {i, j, k} и {x, y, z}. Эти векторы иногда записываются с надчеркиванием (например, ê), чтобы подчеркнуть их статус единичных векторов (стандартных единичных векторов). Эти векторы являются базисом в том смысле, что любой другой вектор может быть единственным образом представлен как их линейная комбинация. Например, каждый вектор v в трехмерном пространстве может быть единственным образом записан как
In mathematics, the standard basis (also called natural basis or canonical basis) of a coordinate vector space (such as or ) is the set of vectors, each of whose components are all zero, except one that equals 1. For example, in the case of the Euclidean plane formed by the pairs (x, y) of real numbers, the standard basis is formed by the vectors
Similarly, the standard basis for the three dimensional space is formed by vectors
Here the vector ex points in the x direction, the vector ey points in the y direction, and the vector ez points in the z direction. There are several common notations for standard basis vectors, including {ex, ey, ez}, {e1, e2, e3}, {i, j, k}, and {x, y, z}. These vectors are sometimes written with a hat to emphasize their status as unit vectors (standard unit vectors). These vectors are a basis in the sense that any other vector can be expressed uniquely as a linear combination of these. For example, every vector v in three dimensional space can be written uniquely as
the scalars , , being the scalar components of the vector v.
In the n dimensional Euclidean space , the standard basis consists of n distinct vectors
where ei denotes the vector with a 1 in the ith coordinate and 0's elsewhere. Standard bases can be defined for other vector spaces, whose definition involves coefficients, such as polynomials and matrices. In both cases, the standard basis consists of the elements of the space such that all coefficients but one are 0 and the non zero one is 1. For polynomials, the standard basis thus consists of the monomials and is commonly called monomial basis. For matrices , the standard basis consists of the m×n matrices with exactly one non zero entry, which is 1. For example, the standard basis for 2×2 matrices is formed by the 4 matrices
, где , , – скалярные компоненты вектора v.
In mathematics, the standard basis (also called natural basis or canonical basis) of a coordinate vector space (such as or ) is the set of vectors, each of whose components are all zero, except one that equals 1. For example, in the case of the Euclidean plane formed by the pairs (x, y) of real numbers, the standard basis is formed by the vectors
Similarly, the standard basis for the three dimensional space is formed by vectors
Here the vector ex points in the x direction, the vector ey points in the y direction, and the vector ez points in the z direction. There are several common notations for standard basis vectors, including {ex, ey, ez}, {e1, e2, e3}, {i, j, k}, and {x, y, z}. These vectors are sometimes written with a hat to emphasize their status as unit vectors (standard unit vectors). These vectors are a basis in the sense that any other vector can be expressed uniquely as a linear combination of these. For example, every vector v in three dimensional space can be written uniquely as
the scalars , , being the scalar components of the vector v.
In the n dimensional Euclidean space , the standard basis consists of n distinct vectors
where ei denotes the vector with a 1 in the ith coordinate and 0's elsewhere. Standard bases can be defined for other vector spaces, whose definition involves coefficients, such as polynomials and matrices. In both cases, the standard basis consists of the elements of the space such that all coefficients but one are 0 and the non zero one is 1. For polynomials, the standard basis thus consists of the monomials and is commonly called monomial basis. For matrices , the standard basis consists of the m×n matrices with exactly one non zero entry, which is 1. For example, the standard basis for 2×2 matrices is formed by the 4 matrices
В n-мерном евклидовом пространстве стандартный базис состоит из n различных векторов
In mathematics, the standard basis (also called natural basis or canonical basis) of a coordinate vector space (such as or ) is the set of vectors, each of whose components are all zero, except one that equals 1. For example, in the case of the Euclidean plane formed by the pairs (x, y) of real numbers, the standard basis is formed by the vectors
Similarly, the standard basis for the three dimensional space is formed by vectors
Here the vector ex points in the x direction, the vector ey points in the y direction, and the vector ez points in the z direction. There are several common notations for standard basis vectors, including {ex, ey, ez}, {e1, e2, e3}, {i, j, k}, and {x, y, z}. These vectors are sometimes written with a hat to emphasize their status as unit vectors (standard unit vectors). These vectors are a basis in the sense that any other vector can be expressed uniquely as a linear combination of these. For example, every vector v in three dimensional space can be written uniquely as
the scalars , , being the scalar components of the vector v.
In the n dimensional Euclidean space , the standard basis consists of n distinct vectors
where ei denotes the vector with a 1 in the ith coordinate and 0's elsewhere. Standard bases can be defined for other vector spaces, whose definition involves coefficients, such as polynomials and matrices. In both cases, the standard basis consists of the elements of the space such that all coefficients but one are 0 and the non zero one is 1. For polynomials, the standard basis thus consists of the monomials and is commonly called monomial basis. For matrices , the standard basis consists of the m×n matrices with exactly one non zero entry, which is 1. For example, the standard basis for 2×2 matrices is formed by the 4 matrices
где обозначает вектор с 1 в i-й координате и 0 в остальных. Стандартные базисы могут быть определены и для других векторных пространств, определение которых включает коэффициенты, такие как полиномы и матрицы. В обоих случаях стандартный базис состоит из элементов пространства, у которых все коэффициенты, кроме одного, равны 0, а ненулевой коэффициент равен 1. Для полиномов стандартный базис таким образом состоит из мономов и обычно называется мономиальным базисом. Для матриц стандартный базис состоит из m×n матриц с ровно одним ненулевым элементом, который равен 1. Например, стандартный базис для матриц 2×2 формируется из 4 матриц
In mathematics, the standard basis (also called natural basis or canonical basis) of a coordinate vector space (such as or ) is the set of vectors, each of whose components are all zero, except one that equals 1. For example, in the case of the Euclidean plane formed by the pairs (x, y) of real numbers, the standard basis is formed by the vectors
Similarly, the standard basis for the three dimensional space is formed by vectors
Here the vector ex points in the x direction, the vector ey points in the y direction, and the vector ez points in the z direction. There are several common notations for standard basis vectors, including {ex, ey, ez}, {e1, e2, e3}, {i, j, k}, and {x, y, z}. These vectors are sometimes written with a hat to emphasize their status as unit vectors (standard unit vectors). These vectors are a basis in the sense that any other vector can be expressed uniquely as a linear combination of these. For example, every vector v in three dimensional space can be written uniquely as
the scalars , , being the scalar components of the vector v.
In the n dimensional Euclidean space , the standard basis consists of n distinct vectors
where ei denotes the vector with a 1 in the ith coordinate and 0's elsewhere. Standard bases can be defined for other vector spaces, whose definition involves coefficients, such as polynomials and matrices. In both cases, the standard basis consists of the elements of the space such that all coefficients but one are 0 and the non zero one is 1. For polynomials, the standard basis thus consists of the monomials and is commonly called monomial basis. For matrices , the standard basis consists of the m×n matrices with exactly one non zero entry, which is 1. For example, the standard basis for 2×2 matrices is formed by the 4 matrices
Свойства
По определению, стандартный базис — это последовательность ортогональных единичных векторов. Иными словами, это упорядоченный и ортонормальный базис. Однако упорядоченный ортонормальный базис не обязательно является стандартным базисом. Например, два вектора, представляющие поворот на 30° 2D стандартного базиса, описанного выше, то есть, также являются ортогональными единичными векторами, но они не совпадают с осями декартовой системы координат, поэтому базис, образованный этими векторами, не соответствует определению стандартного базиса.
are also orthogonal unit vectors, but they are not aligned with the axes of the Cartesian coordinate system, so the basis with these vectors does not meet the definition of standard basis.
Другие виды использования
Существование других "стандартных" базисов стало предметом интереса в алгебраической геометрии, начиная с работ Ходжа 1943 года о грассманианах. В настоящее время это часть теории представлений, известная как стандартная мономиальная теория. Идея стандартного базиса в универсальной обволакивающей алгебре алгебры Ли установлена теоремой Пуанкаре — Биркгоффа — Витта. Базисы Грёбнера также иногда называют стандартными базисами. В физике стандартные базисные векторы для заданного евклидова пространства иногда называют версорами осей соответствующей декартовой системы координат.