Кіріспе
Математикада, координаттық векторлық кеңістіктің стандартты негізі (табиғи негіз немесе канондық негіз деп те аталады) (мысалы, немесе ) – компоненттерінің барлығы 0-ге тең, бірақ біреуі 1-ге тең болатын векторлар жиынтығы. Мысалы, нақты сандардың (x, y) жұптарымен құрылған Евклид жазықтығында стандартты негіз келесі векторлармен құралады:
Vectors whose components are all 0 except one that is 1
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
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
Сонымен қатар, үш өлшемді кеңістік үшін стандартты негіз келесі векторлармен құралады:
Vectors whose components are all 0 except one that is 1
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
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
Мұнда вектор ex x бағытына, вектор ey y бағытына, ал вектор ez z бағытына қарайды. Стандартты негіз векторларын белгілеу үшін бірнеше әдеттегі жазбалар бар, олардың ішінде {ex, ey, ez}, {e1, e2, e3}, {i, j, k} және {x, y, z} жатады. Бұл векторлар кейде бірлік векторлар (стандартты бірлік векторлар) екенін көрсету үшін төбесімен жазылады. Бұл векторлар негіз болып табылады, себебі кез келген басқа векторды осы векторлардың сызықтық комбинациясы түрінде бірегей түрде өрнектеуге болады. Мысалы, үш өлшемді кеңістіктегі кез келген v векторын келесідей бірегей түрде жазуға болады:
Vectors whose components are all 0 except one that is 1
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
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
мұндағы , , – v векторының скалярлық компоненттері.
Vectors whose components are all 0 except one that is 1
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
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
n өлшемді Евклид кеңістігінде стандартты негіз n түрлі вектордан тұрады:
Vectors whose components are all 0 except one that is 1
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
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
мұнда ei – i-інші координатасында 1 және қалғандарында 0 болатын векторды білдіреді. Стандартты негіздерді басқа векторлық кеңістіктер үшін де анықтауға болады, олардың анықтамасы коэффициенттерді, мысалы, көпмүшелер мен матрицаларды қамтиды. Екі жағдайда да стандартты негіз кеңістіктің элементтерінен тұрады, онда барлық коэффициенттердің бірінен басқасы 0-ге тең, ал нөлдік емес коэффициенті 1-ге тең. Көпмүшелер үшін стандартты негіз мономиалдардан тұрады және оны әдетте мономиалдық негіз деп атайды. Матрицалар үшін стандартты негіз дәл бір нөлдік емес элементі бар m×n матрицаларынан тұрады, ол 1-ге тең. Мысалы, 2×2 матрицалар үшін стандартты негіз келесі 4 матрицадан құралады:
Vectors whose components are all 0 except one that is 1
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
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
Қасиеттері
Стандартты негіз – өзара перпендикуляр бірлік векторлардың тізбегі. Басқаша айтқанда, бұл реттелген және ортонормаланған негіз. Дегенмен, реттелген ортонормаланған негіз міндетті түрде стандартты негіз болмайды. Мысалы, жоғарыда сипатталған 2D стандартты негіздің 30° бұрылысын көрсететін екі вектор, яғни,
By definition, the standard basis is a sequence of orthogonal unit vectors. In other words, it is an ordered and orthonormal basis. However, an ordered orthonormal basis is not necessarily a standard basis. For instance the two vectors representing a 30° rotation of the 2D standard basis described above, i. e. ,
де ортогональды бірлік векторлар болып табылады, бірақ олар декарты координаталар жүйесінің өстерімен бірге келмейді, сондықтан осы векторлармен құрылған негіз стандартты негіздің анықтамасына сай келмейді.
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.
Басқа қолданыстар
Басқа "стандартты" негіздердің болуы алгебралық геометрияда қызығушылық тудыратын тақырыпқа айналды, бұл 1943 жылы Ходждің грассмандықтар туралы жұмысынан бастау алды. Қазір бұл стандартты мономиалдық теория деп аталатын бейнелеу теориясының бір бөлігі. Ли алгебрасының әмбебап орауыш алгебрасындағы стандартты негіз идеясы Пуанкаре-Биркхофф-Витт теоремасымен дәлелденеді. Грёбнер негіздері де кейде стандартты негіздер деп аталады. Физикада, белгілі бір евкалид кеңістігінің стандартты негіз векторлары кейде сәйкес картезиандық координаттар жүйесінің осьтерінің версорлары деп аталады.
The existence of other 'standard' bases has become a topic of interest in algebraic geometry, beginning with work of Hodge from 1943 on Grassmannians. It is now a part of representation theory called standard monomial theory. The idea of standard basis in the universal enveloping algebra of a Lie algebra is established by the Poincaré–Birkhoff–Witt theorem. Gröbner bases are also sometimes called standard bases. In physics, the standard basis vectors for a given Euclidean space are sometimes referred to as the versors of the axes of the corresponding Cartesian coordinate system.