Let V={f(x)=c0 + c1x + c2x2 : �ç01 f(x)dx=1}. In other words, V is the set of all polynomials of degree 2 or less such that their integral from 0-1 is equal to 1.
a)Show that the sum of two polynomials in V is not in V
b)Show that an affine combination of two polynomials in V is in V
c)Give a parametric description of the polynomials in V
To understand the properties of the set V, let's break down the problem step by step:
a) Showing that the sum of two polynomials in V is not in V:
To prove this, we need to find two polynomials in V and show that their sum is not in V.
Let's consider two polynomials in V:
1. f(x) = x
2. g(x) = 1 - x
Now, let's find the sum of these two polynomials:
f(x) + g(x) = (x) + (1 - x) = 1
The integral of this sum from 0 to 1 is:
∫[0,1] (f(x) + g(x)) dx = ∫[0,1] 1 dx = [x]^1_0 = 1 - 0 = 1
As you can see, the integral of the sum of f(x) and g(x) is equal to 1, which means the sum is indeed in V. Therefore, the statement "the sum of two polynomials in V is not in V" is incorrect.
b) Showing that an affine combination of two polynomials in V is in V:
To prove this, we need to show that for any two polynomials, p(x) and q(x), in V and any two constants, a and b, the affine combination a*p(x) + b*q(x) is also in V.
Let's consider two polynomials in V:
1. p(x) = c0 + c1x + c2x^2
2. q(x) = d0 + d1x + d2x^2
Now, let's find the affine combination of p(x) and q(x):
a*p(x) + b*q(x) = a*(c0 + c1x + c2x^2) + b*(d0 + d1x + d2x^2)
= (ac0 + bd0) + (ac1 + bd1)x + (ac2 + bd2)x^2
To find the integral of this affine combination from 0 to 1, let's integrate each term:
∫[0,1] (ac0 + bd0) dx = (ac0 + bd0) * [x]^1_0 = (ac0 + bd0) * (1 - 0) = ac0 + bd0
∫[0,1] (ac1 + bd1)x dx = (ac1 + bd1) * [x^2/2]^1_0 = (ac1 + bd1) * (1/2 - 0) = (ac1 + bd1) / 2
∫[0,1] (ac2 + bd2)x^2 dx = (ac2 + bd2) * [x^3/3]^1_0 = (ac2 + bd2) * (1/3 - 0) = (ac2 + bd2) / 3
Adding these integrals together, we get:
(ac0 + bd0) + (ac1 + bd1) / 2 + (ac2 + bd2) / 3
Therefore, the integral of the affine combination is equal to:
(ac0 + bd0) + (ac1 + bd1) / 2 + (ac2 + bd2) / 3
Since this expression is a linear combination of a, b, c0, c1, c2, d0, d1, d2, and the coefficients in front of each term are constants, we can see that the integral of the affine combination is a constant. If this constant is equal to 1, then the affine combination is in V.
c) Parametric description of the polynomials in V:
Let's express the polynomials in V in terms of their coefficients. We know that the integral of each polynomial from 0 to 1 is equal to 1, so we can set up the following equation:
∫[0,1] (c0 + c1x + c2x^2) dx = 1
Integrating the polynomials, we get:
c0[x] + c1[x^2/2] + c2[x^3/3] = 1
Evaluating the integral at 1 and 0, we have:
c0 + c1/2 + c2/3 = 1
Hence, the parametric description of the polynomials in V is given by the equation:
c0 + c1/2 + c2/3 = 1