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In mathematics, a set B of vectors in a vector space V is called a basis (pl.: bases) if every element of V may be written in a unique way as a finite linear combination of elements of B. The coefficients of this linear combination are referred to as components or coordinates of the vector with respect to B .
"A base is a natural number B whose powers (B multiplied by itself some number of times) are specially designated within a numerical system." [1]: 38 The term is not equivalent to radix, as it applies to all numerical notation systems (not just positional ones with a radix) and most systems of spoken numbers. [1]
As the next example will show, present bases are a generalization of the bases of vector spaces. Then, the name "reference frame" can well replace "basis". Yet, contrary to the vector space case, a universal algebra might lack bases and, when it has them, their dimension sets might have different finite positive cardinalities. [6]
In mathematics, exponentiation, denoted b n, is an operation involving two numbers: the base, b, and the exponent or power, n. [1] When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, b n is the product of multiplying n bases: [1] = ⏟.
For example, the Euclidean topology on the plane admits as a base the set of all open rectangles with horizontal and vertical sides, and a nonempty intersection of two such basic open sets is also a basic open set. But another base for the same topology is the collection of all open disks; and here the full (B2) condition is necessary.
The bases of a matroid characterize the matroid completely: a set is independent if and only if it is a subset of a basis. Moreover, one may define a matroid to be a pair (,), where is the ground-set and is a collection of subsets of , called "bases", with the following properties: [7] [8]
Every vector a in three dimensions is a linear combination of the standard basis vectors i, j and k.. 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. [1]
Although the case for two bases, and for d + 1 bases is well studied, very little is known about uncertainty relations for mutually unbiased bases in other circumstances. [ 27 ] [ 28 ] When considering more than two, and less than d + 1 {\displaystyle d+1} bases it is known that large sets of mutually unbiased bases exist which exhibit very ...