Question

Show that the set P(n) of all polynomials of degree less than or equal to n is a subspace of C(1). What is dim P(2)? What is dim P(n)?

Answer 1

\[\dim(p _{2})=3\] with its natural basis {1,x,x^2} and \[\dim(p _{n})=n+1\]

Answer 2

so how do i show the set P(n) mentioned is a subspace of C(1)?

Answer 3

to show that P(n) is a subspace, we must show that it is closed under addition and scalar multiplication,

Answer 4

do u understand @zonazoo

Answer 5

yes, I do understand that, but I do not know exactly how to do show that.

Answer 6

we know the following two axioms which you have to prove:

1.for all \[\alpha,\beta \in C\], \[\alpha+\beta \in C\]
2. \[\forall \alpha \in C,\exists a \in F\] such that \[a \alpha \in C\], you do the same with polynomials, define polynomial 1 and polynomial2 like i did for alpha and beta

Answer 7

try it, oh and we know that if any of the axioms fails its not a subspace but with our case they know that its a subspace u only have to show , and show me what u get

Answer 8

I just doesn't get where to start since there are no polynomials given.

Answer 9

you let \[P _{1}=a _{0}+a _{1}x+a_{2}x^{2}+...+a _{n}x ^{n}\] and \[P _{2}=b _{0}+b _{1}x+b _{2}x ^{2}+...+b _{n}x ^{n}\] and prove the axioms of a subspace