Chapter 3.1, Problem 16E

We can define a one−to−one correspondence between the elements of $P_n$ and ${ℝ}^n$ by
$p\left(x\right)=a_1+a_{2}x+\mathrm{...}+a_n{x}^{n-1}\leftrightarrow {\left(a_1,\mathrm{...},a_n\right)}^T=\text{a}$
Show that if $p\leftrightarrow \text{a}$ and $q\leftrightarrow \text{b}$ , then
(a) $\alpha p\leftrightarrow \alpha \text{a}$ for any scalar $\alpha$ .
(b) $p+q\leftrightarrow \text{a+b}$ .
[In general, two vector spaces are said to be isomorphic if their elements can be put into a one−to−one correspondence that is preserved under scalar multiplication and addition as in (a) and (b).]