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A projective basis is + points in general position, in a projective space of dimension n. A convex basis of a polytope is the set of the vertices of its convex hull. A cone basis [5] consists of one point by edge of a polygonal cone. See also a Hilbert basis (linear programming).
In mathematics, a canonical basis is a basis of an algebraic structure that is canonical in a sense that depends on the precise context: In a coordinate space , and more generally in a free module , it refers to the standard basis defined by the Kronecker delta .
A basis (or reference frame) of a (universal) algebra is a function that takes some algebra elements as values () and satisfies either one of the following two equivalent conditions. Here, the set of all b ( i ) {\displaystyle b(i)} is called the basis set , whereas several authors call it the "basis".
In linear algebra, given a vector space with a basis of vectors indexed by an index set (the cardinality of is the dimension of ), the dual set of is a set of vectors in the dual space with the same index set such that and form a biorthogonal system.
In mathematics, a basis function is an element of a particular basis for a function space. Every function in the function space can be represented as a linear combination of basis functions, just as every vector in a vector space can be represented as a linear combination of basis vectors .
is the change-of-basis matrix (also called transition matrix), which is the matrix whose columns are the coordinates of the new basis vectors on the old basis. A change of basis is sometimes called a change of coordinates, although it excludes many coordinate transformations.
In mathematics, particularly linear algebra, an orthogonal basis for an inner product space is a basis for whose vectors are mutually orthogonal. If the vectors of an orthogonal basis are normalized , the resulting basis is an orthonormal basis .
A basis formed this way is called a standard basis for the geometric algebra, and any other orthogonal basis for will produce another standard basis. Each standard basis consists of elements. Every multivector of the geometric algebra can be expressed as a linear combination of the standard basis elements.