Question

Consider ℙ2 with the inner product given by evaluation at −1​, ​0, and 1. Compute ||p|| and ||q|| for ​p(t)=2t−t2 and ​q(t)=2+5t^2.

Answer 1

The norms of the polynomials p(t) and q(t) are computed by evaluating at points -1, 0, and 1. We find that ||p|| = √(10) and ||q|| = √(98).

Step-by-step explanation:

The question asks us to compute the norms (or magnitudes) of two polynomials treated as vectors, with an inner product defined by evaluation at the points -1, 0, and 1.

The polynomials in question are p(t) = 2t - t^2 and q(t) = 2 + 5t^2.

The norm of a polynomial p, denoted ||p||, in this inner product space is defined as the square root of the inner product of the polynomial with itself, which can be calculated by evaluating the polynomial at the given points, squaring the results, and adding them together.

For p(t), we evaluate at the points -1, 0, and 1

• p(-1) = 2(-1) - (-1)^2 = -2 - 1 = -3
• p(0) = 2(0) - (0)^2 = 0
• p(1) = 2(1) - (1)^2 = 2 - 1 = 1

The norm ||p|| = √((p(-1))^2 + (p(0))^2 + (p(1))^2)

= √((-3)^2 + (0)^2 + (1)^2)

= √(10).

For q(t), we evaluate at the points -1, 0, and 1:

• q(-1) = 2 + 5(-1)^2 = 2 + 5 = 7
• q(0) = 2 + 5(0)^2 = 2
• q(1) = 2 + 5(1)^2 = 2 + 5 = 7

The norm ||q|| = √((q(-1))^2 + (q(0))^2 + (q(1))^2)

= √((7)^2 + (2)^2 + (7)^2)

= √(98).