Standard basis

In linear algebra, the standard basis or canonical basis for the n-dimensional coordinate space is the basis obtained by taking the n basis vectors { e_j: 1leq jleq n} where e_j is the vector with a 1 in the jth coordinate and 0 elsewhere. In many senses, it is the "obvious" basis.

Related Topics:
Linear algebra - Dimension - Coordinate space - Basis - Coordinate

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Standard basis are perfectly localized in the sense that all but one element of each base are zero.

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There is a standard basis also for the ring of polynomials in n indeterminates over a field, namely the monomials.

Related Topics:
Polynomial - Field - Monomial

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All of the preceding are special cases of the family

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:{(e_i)}_{iin I}={({(delta_{ij})}_{jin I})}_{iin I}

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where I is any set and delta_{ij} is the Kronecker delta, equal to zero whenever i≠j and equal to 1 if i=j.

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This family is the canonical basis of the R-module (free module)

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:R^{(I)}

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of all families

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:f=(f_i)

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from I into a ring R, which are zero except for a finite number of indices, if we interpret 1 as 1R, the unit in R.

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