Is Every Subset a Subspace in Vector Spaces of Functions?

[NeroBlade]

Hi

How can I work out which subset is a subspace and which one isn't on this problem:

Fq ([0,1]) be vector space of all functions [0,1] -> Q with addition and scalar multiplication defined in usual way.

Let U < Fq ([0,1]) be the subset consisting of all functions f s.t. f(0) >= f(1) and let
V < Fq([0,1]) be subset consist of all functions f such that f(0) = f(1).

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And lastly if C2 is a complex vector space consisting of pairs (z1 and z2) of complex numbers what's the 3 different bases for (Complex numbers base 2)?

 

[HallsofIvy]

Hi

How can I work out which subset is a subspace and which one isn't on this problem:

Fq ([0,1]) be vector space of all functions [0,1] -> Q with addition and scalar multiplication defined in usual way.

Let U < Fq ([0,1]) be the subset consisting of all functions f s.t. f(0) >= f(1) and let
V < Fq([0,1]) be subset consist of all functions f such that f(0) = f(1).

===

To prove that "U is a subspace of V" you must only prove that, for any u, v in U, au+bv is int U which is equivalent to proving "if u and v are in U, then u+v is in U" and "if u is in U and a is a scalar, then au is int U'.

And lastly if C2 is a complex vector space consisting of pairs (z1 and z2) of complex numbers what's the 3 different bases for (Complex numbers base 2)?

I have no idea what you mean by this. There exit an infinite number of bases for any vector space. Nor do I understand what you mean by "Complex numbers base 2". Do you mean "modulo 2"?