I've attached an image of my problem set. I'm confused about what they mean to define the operation of addition or scalar multiplication, do they mean which of the eight axioms? Thank you.
You need to show that all of the vectorspace axioms hold.
Identity There exists a Vector $\displaystyle \vec{0}$ such that
$\displaystyle \vec{0}+\vec u =\vec u$ for all vectors u
in this vector space 1 is the additive identity.
let u be any vector in the vector space then
$\displaystyle 1+u=1 \cdot u=u \iff 1+u=u$
So this axiom holds.
Try to verify all of the other Axioms.
let k be a scalar
Distibutive Axiom:
$\displaystyle k(\vec u + \vec v)=k\vec u + k\vec v$
now useing the def on operations we need to show that this holds so
$\displaystyle k(\vec u + \vec v)=\underbrace{k(u\cdot v)}_{\mbox{Vector add}}= \underbrace{(uv)^k}_{\mbox{ Scalar mult}}=u^kv^k=k\vec u + k\vec v$
The the distributive law of scalar mult holds.