1. Clarification on a problem

Hey guys,
I have this problem: Prove that U1 := {(x1,...,xn) ∈ F^n | x1 +...+xn = 0} is a subspace of F^n, but U2 := {(x1, . . . , xn) ∈ F^n | x1 + . . . + xn = 1} is not.

My question is do I go about showing U1 as a subspace and U2 not as a subspace as one would do normally? Or are there special conditions we have to follow because we are dealing with F^n (n-tuples). I tried looking in my book and my book doesn't really show anything on it and I tried google but i can't seem to find what i'm looking for.
Thank you.

2. Originally Posted by alice8675309
Hey guys,
I have this problem: Prove that U1 := {(x1,...,xn) ∈ F^n | x1 +...+xn = 0} is a subspace of F^n, but U2 := {(x1, . . . , xn) ∈ F^n | x1 + . . . + xn = 1} is not.

My question is do I go about showing U1 as a subspace and U2 not as a subspace as one would do normally? Or are there special conditions we have to follow because we are dealing with F^n (n-tuples). I tried looking in my book and my book doesn't really show anything on it and I tried google but i can't seem to find what i'm looking for.
Thank you.

Show that $U_1$ isn't empty and is closed under vector addition and multiplication by scalars, and then show

either that $U_2$ isn't closed under neither of both operations above (pretty easy to do), or else show it doesn't

contain the zero vector (why is this a necessary, though not sufficient, condition for something to be a subspace?)

Tonio

3. Originally Posted by tonio
Show that $U_1$ isn't empty and is closed under vector addition and multiplication by scalars, and then show

either that $U_2$ isn't closed under neither of both operations above (pretty easy to do), or else show it doesn't

contain the zero vector (why is this a necessary, though not sufficient, condition for something to be a subspace?)

Tonio
ok got it. Thanks! I was just making sure that it was the same usual steps and not something different because we were dealing with n-tuples. Couldn't remember. Thanks again