I have difficulty understand how to proof if something is vector space. I miss lecture as I was sick and the book is not help a lot. Here is the question
Let $P$ be the set of all polynomials. Show that $P$, together with the usual addition and scalar multiplication of functions, forms a vector space.
I understand that we should use the 8 axioms to help us but how I would go about using those if the question is asking very generally. Any help is good help
P.S Pardon my English
EDIT: The 8 axioms are:
- $x + y = y + x$ for any $x$ and $y$ in $V$.
- $(x + y) + z = x + (y + z)$ for any $x$, $y$, and $z$ in $V$.
- There exists an element $0$ in $V$ such that $x + 0 = x$ for each $x \in V$.
- For each $x \in V$, there exists an element $−x$ in $V$ such that $x + (−x) = 0.$
- $α(x + y) = αx + αy$ for each scalar $α$ and any $x$ and $y$ in $V$.
- $(α + β)x = αx + βx$ for any scalars $α$ and $β$ and any $x \in V$.
- $(αβ)x = α(βx)$ for any scalars $α$ and $β$ and any $x \in V$.
- $1 · x = x$ for all $x \in V$.