- #1

Cassi

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## Homework Statement

Let P

_{n}denote the linear space of all real polynomials of degree </= n, where n is fixed. Let S denote the set of all polynomials f in P

_{n}satisfying the condition given. Determine whether or not S is a subspace of P

_{n}. If S is a subspace, compute dim S.

The given condition if f(0)=f(1)

## Homework Equations

Closure axioms:

(1)Closure under addition: For every pair of elements x and y in V there corresponds a unique element in V called the sum of x and y denoted by x+y.

(2) Closure under multiplication: For every x and y in V and every real number a there corresponds an element in V called the product of a and x, denoted by ax.

## The Attempt at a Solution

I[/B] know that S is a subspace of P

_{n}if S is a subset of P

_{n}and it satisfies the closure axioms. So I need to prove these three things, but they seem trivial to me.

S c Pn because S: f(0)=f(1) has all elements in Pn. Is this enough of a proof to draw the conclusion?

For the closure axioms, I know that I have to show that f(x)+g(x) is an element in Pn, but again this seems trivial, I am unsure how to do this. Same for the closure axiom of multiplication.

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