Proving that Polynomials of Degree Less Than or Equal to n Form a Vector Space

In order to prove that the set of polynomials of degree less than or equal to N, denoted as P_N, forms a vector space over the real numbers mathbb{R}, we need to demonstrate that P_N satisfies all the vector space axioms.

Step 1: Define the Set P_N

Let P_N be the set of all polynomials of degree less than or equal to N:

P_N { p(x) a_N x^N a_{N-1} x^{N-1} ldots a_1 x a_0 mid a_i in mathbb{R}, N in mathbb{N} }

Step 2: Check Closure Under Addition

Take two polynomials p(x) in P_N and q(x) in P_N:

p(x) a_N x^N a_{N-1} x^{N-1} ldots a_1 x a_0

q(x) b_N x^N b_{N-1} x^{N-1} ldots b_1 x b_0

Now consider the sum r(x) p(x) q(x):

r(x) (a_N b_N)x^N (a_{N-1} b_{N-1})x^{N-1} ldots (a_1 b_1)x (a_0 b_0)

The highest degree term in r(x) is (a_N b_N)x^N. If both a_N and b_N are zero, then the degree of r(x) is less than N. Otherwise, the degree of r(x) is still N. Thus, r(x) in P_N.

Step 3: Check Closure Under Scalar Multiplication

Take a polynomial p(x) in P_N and a scalar c in mathbb{R}:

p(x) a_N x^N a_{N-1} x^{N-1} ldots a_1 x a_0

Now consider the scalar multiplication c cdot p(x):

c cdot p(x) c(a_N x^N a_{N-1} x^{N-1} ldots a_1 x a_0) c a_N x^N c a_{N-1} x^{N-1} ldots c a_1 x c a_0

The highest degree term is c a_N x^N. Similar to before, if a_N 0, then the degree of the polynomial will be less than N. Otherwise, it will remain N. Thus, c cdot p(x) in P_N.

Step 4: Verify Vector Space Axioms

Now that we have shown closure under addition and scalar multiplication, we need to verify the vector space axioms:

Associativity of Addition

For all p(x), q(x), r(x) in P_N,

p(x) (q(x) r(x)) (p(x) q(x)) r(x)

Commutativity of Addition

For all p(x), q(x) in P_N,

p(x) q(x) q(x) p(x)

Existence of Additive Identity

The zero polynomial 0 in P_N such that for all p(x) in P_N,

p(x) 0 p(x)

Existence of Additive Inverses

For each p(x) in P_N, there exists -p(x) in P_N such that

p(x) (-p(x)) 0

Distributive Properties

c(p(x) q(x)) c p(x) c q(x)

(c d)p(x) c p(x) d p(x)

Associativity of Scalar Multiplication

For all c, d in mathbb{R},

(c d) p(x) c(d p(x))

Identity Element of Scalar Multiplication

For all p(x) in P_N,

1 cdot p(x) p(x)

Conclusion

Since P_N satisfies all the vector space axioms, we conclude that the set of polynomials of degree less than or equal to N forms a vector space over mathbb{R}.