# Linear Algebra - Vector Spaces

Let P be the set of all polynomials. Show that P, with the usual addition and scalar multiplication of functions, forms a vector space.

I'm just no good at proofs. I know we are supposed to go through and prove the Vector Space Axioms and the C1 and C2 closure properties. I just don't think I'm doing it successfully. I'm just not understanding this. For instance, the closure properties don't seem very different from a couple of the axioms.

Axioms:

A1. x + y = y + x for any x and y in V.

A2. (x + y) + z = x + (y + z) for any x,y,z in V.

A3. There exists an element 0 in V such that x + 0 = x for each x in the set V.

A4. For each x in the set V, there exists an element -x in V such that x + (-x) = 0.

A5. alpha(x + y) = alpha*x + alpha*y for each scalar alpha and any x and y in V.

A6. (alpha + beta)x = alpha*x + beta*x for any scalars alpha and beta and any x that belongs to the set V.

A7. (alpha*beta)x = alpha(beta*x) for any scalars alpha and beta and any x that belongs to the set V.

A8. 1*x = x for all x in V.

Closure properties:

C1: If x is in V and alpha is a scalar, then alpha*x is in V.

C2. If x,y is in V, then x + y is in V.

#### Solution Preview

Proof: I want to show P is a vector space, I have to go through all the Axioms and Closure properties. Suppose p1,p2,p3 are three arbitrary polynomials in P.

<br> p1=Dnx^n+...+D1x+D0, p2=Bmx^m+...+B1x+B0 and suppose n>=m without the loss ...