What Is a Basis for Vector Spaces of Finite Nonzero Term Sequences?

[eckiller]

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.

 

[matt grime]

e(n), which is zero for i=/=n, 1 for i=n is a basis.

 

[mathwonk]

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...").