Inner Product Examples for Polynomial Spaces Pn with Worked Solutions

📚 Quick Study Guide

• 🔢 Definition: An inner product on a polynomial space $P_n$ is a function that takes two polynomials $p(x)$ and $q(x)$ and returns a scalar, satisfying certain axioms (linearity, symmetry, and positive-definiteness).
• ➕ Standard Inner Product: A common inner product is defined as $\langle p, q \rangle = \int_a^b p(x)q(x) dx$, where $[a, b]$ is an interval.
• ⚖️ Weighted Inner Product: Another type is the weighted inner product: $\langle p, q \rangle = \int_a^b w(x)p(x)q(x) dx$, where $w(x)$ is a weight function (positive on $[a, b]$).
• 📏 Discrete Inner Product: Given points $x_0, x_1, ..., x_n$, the discrete inner product is $\langle p, q \rangle = \sum_{i=0}^n p(x_i)q(x_i)$.
• 💡 Orthogonality: Two polynomials $p(x)$ and $q(x)$ are orthogonal if their inner product is zero, i.e., $\langle p, q \rangle = 0$.
• 📝 Norm: The norm (or length) of a polynomial $p(x)$ is defined as $||p|| = \sqrt{\langle p, p \rangle}$.

Practice Quiz

1. Question 1: Which of the following is a valid inner product for polynomials $p(x)$ and $q(x)$ in $P_n$?
   1. $\langle p, q \rangle = p(0) + q(0)$
   2. $\langle p, q \rangle = p(1)q(1) + p(2)q(2)$
   3. $\langle p, q \rangle = \int_{-1}^{0} p(x)q(x) dx$
   4. $\langle p, q \rangle = p'(0)q'(0)$
2. Question 2: Consider the inner product $\langle p, q \rangle = \int_0^1 p(x)q(x) dx$ on $P_1$. If $p(x) = x$ and $q(x) = x + 1$, what is $\langle p, q \rangle$?
   1. $\frac{1}{2}$
   2. $\frac{5}{6}$
   3. $\frac{2}{3}$
   4. $1$
3. Question 3: Which condition must a weight function $w(x)$ satisfy for $\langle p, q \rangle = \int_a^b w(x)p(x)q(x) dx$ to be a valid inner product?
   1. $w(x) < 0$ for all $x$ in $[a, b]$
   2. $w(x) = 0$ for some $x$ in $[a, b]$
   3. $w(x) > 0$ for all $x$ in $[a, b]$
   4. $w(x)$ is a constant function
4. Question 4: Let $\langle p, q \rangle = p(0)q(0) + p(1)q(1)$ be an inner product on $P_1$. Are $p(x) = x$ and $q(x) = 1-x$ orthogonal?
   1. Yes
   2. No
   3. Cannot be determined
   4. Only if $x = 0$
5. Question 5: For the inner product $\langle p, q \rangle = \int_0^1 p(x)q(x) dx$, find the norm of $p(x) = x$.
   1. $\frac{1}{2}$
   2. $\frac{1}{\sqrt{3}}$
   3. $\frac{1}{3}$
   4. $\frac{1}{\sqrt{2}}$
6. Question 6: Suppose $\langle p, q \rangle = \int_{-1}^1 p(x)q(x) dx$. Which of the following polynomial pairs are orthogonal?
   1. $p(x) = x, q(x) = x^2$
   2. $p(x) = 1, q(x) = x+1$
   3. $p(x) = x, q(x) = x+1$
   4. $p(x) = 1, q(x) = x^2$
7. Question 7: Which of the following statements is always true for an inner product $\langle \cdot, \cdot \rangle$ on $P_n$?
   1. $\langle p, p \rangle < 0$ for all non-zero $p$
   2. $\langle p, q \rangle = -\langle q, p \rangle$
   3. $\langle p, p \rangle \geq 0$ for all $p$
   4. $\langle 0, p \rangle = 1$ for all $p$