Question

Linear algebra question

Answer 1

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...

Answer 2

Associativity looks good...

Answer 3

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.