Question: We can define a one-to-one correspondence between the elements of Pn and Rn by p(x) = a1 + a2x + . . . + anxn-1

We can define a one-to-one correspondence between the elements of Pn and Rn by
p(x) = a1 + a2x + . . . + anxn-1 ↔ (a1, . . . , an)T = a
Show that if p ↔ a and q ↔ b, then
(a) αp ↔ αa for any scalar a.
(b) p + q ↔ 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).]

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