Re: What is a basis for vector space of {(a_1,a_2,...)} a_i real?
From: Stephen J. Herschkorn (herschko_at_rutcor.rutgers.edu)
Date: 08/30/04
- Next message: Stephen J. Herschkorn: "Re: What is a basis for vector space of {(a_1,a_2,...)} a_i real?"
- Previous message: Peter Webb: "Re: Set theory question ...."
- In reply to: N. Silver: "Re: What is a basis for vector space of {(a_1,a_2,...)} a_i real?"
- Next in thread: Stephen J. Herschkorn: "Re: What is a basis for vector space of {(a_1,a_2,...)} a_i real?"
- Reply: Stephen J. Herschkorn: "Re: What is a basis for vector space of {(a_1,a_2,...)} a_i real?"
- Reply: N. Silver: "Re: What is a basis for vector space of {(a_1,a_2,...)} a_i real?"
- Messages sorted by: [ date ] [ thread ]
Date: Mon, 30 Aug 2004 06:04:27 GMT
N. Silver wrote:
>Ron Jones wrote:
>
>
>
>>I'd be very grateful if anyone could answer the following.
>>If V is the set of all countable-tuples (a_1,a_2,a_3,...)
>>where a_i are rational or real, then under usual pointwise
>>addition and scalar multiplication it is a vector space.
>>Every vector space has a basis - what then is a basis of V?
>>
>>
>
>The standard basis is:
>e1 = (1,0,0,...),
>e2 = (0,1,0,...),
>e3 = (0,0,1,...),...
>
>Then a vector (a_1,a_2,a_3,...) = (a_1)e1 + (a_2)e2 +...
>
>This space is isomorphic to the space of polynomials,
>however you want to choose the coefficients (either only
>rationals or real coefficients). The standard basis for this
>space is {1, x, x^2, x^3,...}.
>
Wrong. How would you express the vector (1,1,1,...) as a finite linear
combination of elements of this "standard basis"?
By definition, a subset B of a vector space V over a field F iff
for every v in V, there exist *finite* subsets {f1, f2,.., f_n} of
F and {b1, b2,..., b_n} of B such that v = sum (i = 1..n, f_i b_i).
-- Stephen J. Herschkorn herschko@rutcor.rutgers.edu
- Next message: Stephen J. Herschkorn: "Re: What is a basis for vector space of {(a_1,a_2,...)} a_i real?"
- Previous message: Peter Webb: "Re: Set theory question ...."
- In reply to: N. Silver: "Re: What is a basis for vector space of {(a_1,a_2,...)} a_i real?"
- Next in thread: Stephen J. Herschkorn: "Re: What is a basis for vector space of {(a_1,a_2,...)} a_i real?"
- Reply: Stephen J. Herschkorn: "Re: What is a basis for vector space of {(a_1,a_2,...)} a_i real?"
- Reply: N. Silver: "Re: What is a basis for vector space of {(a_1,a_2,...)} a_i real?"
- Messages sorted by: [ date ] [ thread ]
Relevant Pages
|
