Vector space: Is A1 = {g belongs to A|g(2) = 3g(4)} a vector space ?

[bluemath]

Hello,

We have A = {g belongs to Functions(R,R) | g(x) = a₀ + a₁x + a₂x^2 + a₃x^3, with a_i belongs to R} and A is a vector space.

Is A₁ = {g belongs to A | g(2) = 3g(4)} a vector space ?
Same question for A2 = {g belongs to A | g(x) >= 0 for all x belongs to [-1;1]} ?

Thanks in advance for some help.

 

[Ishuda]

Hello,

We have A = {g belongs to Functions(R,R) | g(x) = a₀ + a₁x + a₂x^2 + a₃x^3, with a_i belongs to R}

Is A₁ = {g belongs to A | g(2) = 3g(4)} a vector space ?
Same question for A2 = {g belongs to A | g(x) >= 0 for all x belongs to [-1;1]} ?

Thanks in advance for some help.

What are your thoughts? What have you done so far? Please show us your work even if you feel that it is wrong so we may try to help you. You might also read
http://www.freemathhelp.com/forum/threads/78006-Read-Before-Posting

If you need help in determining what a vector space is [the axioms which must be satisfied], you might look at
https://en.wikipedia.org/wiki/Vector_space

 

[bluemath]

Thanks for your answer.

I did a misstake and put a correction in my first post : A is vector space.

About A1 :

g(2) = 3g(4)

and now I don't understand how to test the axioms with that.
Probably A1 is not a vector space and it will be more easy to find a counter example but I'm troubled by this g(2) = 3g(4)