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

In summary, the basis for the vector space V, which consists of all sequences g(n) = a_n in F with a finite number of nonzero terms, is the sequence of monomials 1, X, X^2, X^3, etc. This is equivalent to the space of polynomials, as a sequence of coefficients is the same as a polynomial.
  • #1
eckiller
44
0
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.
 
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  • #2
e(n), which is zero for i=/=n, 1 for i=n is a basis.
 
  • #3
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...").
 

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

What is a basis in linear algebra?

A basis in linear algebra is a set of linearly independent vectors that span a vector space. This means that any vector in the space can be expressed as a linear combination of the basis vectors.

How do you determine if a set of vectors is a basis?

To determine if a set of vectors is a basis, you can use the linear dependence test. If the vectors are linearly independent, meaning that none of the vectors can be written as a linear combination of the others, then the set is a basis.

Can a vector space have more than one basis?

Yes, a vector space can have multiple bases. In fact, any vector space with dimension greater than 1 will have infinitely many possible bases.

What is the difference between a basis and a spanning set?

A basis is a specific type of spanning set that is both linearly independent and spans the entire vector space. A spanning set, on the other hand, can be any set of vectors that spans the space, but may not be linearly independent.

How is the dimension of a vector space related to its basis?

The dimension of a vector space is equal to the number of vectors in its basis. This means that every basis for a given vector space will have the same number of vectors, and any set of linearly independent vectors with that same number can be used as a basis.

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