Question

Let p(x) = a1x^2 + b1x +c1 and q(x) = a2x^2 + b2x + c2 be polynomials in P2. Define an inner product in P2 as follows {p,q} = 5a1a2 + 4b1b2 + 3c1c2.

Given p(x) =5x^2 + (-1)x + (-3) and q(x) = 2x^2 + (4)x +(-3). Evaluate the following expressions
1. p(x) - q(x) = 3x^2 - 5x
2. {p - q, p-q} = 145
3. llp-qll = sqrt({p-q,p-q}) = sqrt(145)

For part 1, I know the answer and how to get it.
For part 2, I know the answer but I'm not sure how to get to it

Answer 1

Explanation:

To evaluate the expression {p - q, p - q}, which represents the inner product of the polynomial (p - q) with itself, you can follow these steps:

Given p(x) = 5x^2 - x - 3 and q(x) = 2x^2 + 4x - 3.

Subtract q(x) from p(x) to get (p - q):

(p - q)(x) = (5x^2 - x - 3) - (2x^2 + 4x - 3)

= 5x^2 - x - 3 - 2x^2 - 4x + 3

= (5x^2 - 2x^2) + (-x - 4x) + (-3 + 3)

= 3x^2 - 5x

Now, calculate the inner product of (p - q) with itself using the given inner product formula:

{p - q, p - q} = 5(a1)(a2) + 4(b1)(b2) + 3(c1)(c2)

= 5(3)(3) + 4(-5)(-5) + 3(0)(0)

= 45 + 100 + 0

= 145

Therefore, the value of {p - q, p - q} is 145.