3 ms·
> any given vector of the space can be represented using a linear combination of a finite subset of the basis elements with rational scalar coefficients? FWIW,
by a1369209993 6y ago
> any given vector of the space can be represented using a linear combination of a finite subset of the basis elements with rational scalar coefficients?
FWIW, this is definitely false:
x = q₀·√2/1 + q₁·√3/3 + q₂·√5/9 + q₃·√7/27 + q₄·√11/81 + ... = Σᵢ(qᵢ·√πᵢ/3ⁱ)
for rational qᵢ.
Edit: primes approximate e^i, so 2^i isn't large enough.
- deleted 6y ago[deleted]
- fractionalhare 6y agoThis isn't false; it's a theorem. Using the axiom of choice, all vectors of any space can be represented as finite linear combinations. In the case of infinite-dimensional vector spaces, this means you would need at most n vectors from the infinite basis B, and n scalars from the underlying field F. I'm interested in what you're trying to demonstrate with your example though, or if we misunderstand each other, because this theorem is very well established.