## anonymous 5 years ago Linear algebra question

1. anonymous

Sorry if this seems like a double post, I added it to the end of a question I marked as answered, so I figured it would never get read... The question asks you to show that the set $\Pi _{n}=\left\{ p:[0,1]\rightarrow \mathbb{R} | \sum_{n}^{j=0} a _{j}x ^{j}, a _{0},...,a_{n} \in \mathbb{R}\right\}$ equipped with pointwise addition and scalar multiplication is a vector space, by verifying the 8 axioms of vector spaces. Oh and $n \in \mathbb{N}$ So with the associativity of addition axiom, I would need to show: $p(x),q(x),r(x) \in \Pi_n, (p+(q+r))(x) = ((p+q)+r)(x)$ (Correct me if I'm wrong) Would I do this by saying that, due to pointwise addition, and because p(x),q(x) and r(x) are real numbers (hence addition of p,q,r is associative) $(p+(q+r))(x) = p(x)+(q(x)+r(x)) = p(x)+q(x)+r(x)$ $= (p(x)+q(x))+r(x) = ((p+q)+r)(x)$ I can't seem to get my head round this sort of maths. I never know how rigorous the proofs have to be either...

2. anonymous

Associativity looks good...

3. anonymous

Oh fab thanks. I really never know if what I've written is enough/correct. I prefer maths when the right answer is obvious when you've actually got to it.