Recent content by astral

Excellent. Thank you very much for your help.

Thanks for your patience, and I am definitely getting a better understanding of this now. I'm just new to vector spaces, and I just want to be sure I know what I can assume when dealing with these problems. I think I'll completely understand if you show me how you would prove that an addition...

hmmmm. It just seems to me like your method is simply proving commutativity by assuming commutativity, which intrinsically doesn't prove anything. maybe I just don't understand.

I thought of doing it this way, which seems a bit better: (f + g)(s) = f(s) + g(s) (g + f)(s) = g(s) + f(s) (f + g)(s) + f(s) = ( f(s) + g(s) ) + f(s) [by the definition] (f + g)(s) + f(s) = f(s) + ( g(s) + f(s) ) [by the associativity axiom] (f + g)(s) + f(s) = f(s) + (g + f)(s) (by...

I'm sorry if the answer is really obvious, but could you give me an example anyway? Thanks.

Let V denote the set of all differentiable real-valued functions defined on the real line. Prove that V is a vector space with the operations of addition and scalar multiplication as follows: (f + g)(s) = f(s) + g(s) and (cf)(s) = c[f(s)] --------------- I know I have to prove this by...