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) p a 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 t...

If px a 1 a 2 x anxn1 a a 1 a 2 an T qx b 1 b 2 x bnxn ...

We can define a one-to-one correspondence between the elements of Pn and Rn by p(x) = a1

Question:

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).]