I have just started learning dual spaces and in my current understanding, I know that dual vectors are vectors that maps vectors onto scalars. However, how do I prove if a vector is a dual vector? For example, let V be the space of real polynomials of degree < 3.
Are the following dual vectors?
- p → p(3) p'(4)
- p → ev5 ((x+1)p(x))
If dual vectors maps vectors onto scalars, does this mean that both 1 and 2 are considered dual vectors? If not, how do I prove that they are not dual vectors?
Any help would be greatly appreciated!
Dual vectors are linear maps from the "original" to the field of scalars $\mathbb F$, let's take the first map, $F(p)=p(3)p'(4)$ with $p \in \mathbb F_2[x]$ .
Well, $F$ it's a map from $\mathbb F_2[x] $ to $\mathbb F$, so what is left to check is the linearity, therefore we have to check that:
$\forall p,q \in F_2[x]$ , $F(p+q)=F(p)+F(q)$
$\forall p \in F_2[x]$ and $\forall a\in \mathbb F$ , $F(ap)=aF(p)$
Can you continue from here?