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Linear algebra basis question

  1. Feb 3, 2005 #1
    What is a basis for the vector space V which consists of all sequences

    g(n) = a_n

    in F that have only a finite number of nonzero terms a_n.

    (Def: A sequence in F is a function g from the positive integers into F).

    I don't know, I can "see" euclidean, polynomial, and matrix bases in my head, but not function and sequence bases.

    Please explain so that I can learn. Thanks in advanced.
     
  2. jcsd
  3. Feb 3, 2005 #2

    matt grime

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    e(n), which is zero for i=/=n, 1 for i=n is a basis.
     
  4. Feb 3, 2005 #3

    mathwonk

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    your space is the same as the space of all polynomials, i.e. a finite sequence of elements of F, is just the sequence of coefficients of some polynomial.

    so as Matt said, the natural basis is the sequence of monomials: 1, X, X^2, X^3,.....

    i point this out since you said you liked polyonmials better than sequences. actually there is no difference. in fact the rigorous definition of a polynomial is as a sequence of coefficients (rather than "an expression of form...").
     
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