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Let $V$ be a vector space. We say that the inner product of vectors $\vec{v_1}$ and $\vec{v_2}$, written $\langle v_1,v_2 \rangle$, is a function that takes any two vectors to a number in $\mathbb{F}$ and has the following additional properties:
1. (Symmetry) :
$$\langle x,y \rangle = \langle y,x \rangle,$$
2. (Linearity in the first argument) The following formula holds for all scalars $\alpha,\beta$ and vectors $\vec{x},\vec{y},\vec{z}$:
$$\langle \alpha \vec{x}+\beta \vec{y},\vec{z} \rangle = \alpha \langle \vec{x},\vec{z} \rangle + \beta \langle \vec{y},\vec{z} \rangle.$$
3. (Positive definiteness) It is always true that
$$\langle \vec{x},\vec{x} \rangle \geq 0$$
and $\langle \vec{x},\vec{x} \rangle=0$ if and only if $\vec{x}=\vec{0}$.
If $V$ is a vector space and $\langle \cdot,\cdot \rangle$ is an inner product on $V$, then we say that $(V,\langle \cdot,\cdot \rangle)$ is an inner product space.
Examples of inner product spaces
1. Let $\vec{x},\vec{y} \in \mathbb{R}^{n \times 1}$ with $\vec{x}=\begin{bmatrix} x_1 \\ \vdots \\ x_n \end{bmatrix}$ and $\vec{y}=\begin{bmatrix} y_1 \\ \vdots \\ y_n \end{bmatrix}$. Then the following formula defines an inner product:
$$\langle \vec{x}, \vec{y} \rangle = x_1y_1 + x_2y_2 + \ldots + x_ny_n = \displaystyle\sum_{k=1}^n x_ky_k.$$
2. Let $\vec{p},\vec{q} \in \mathcal{P}$ (the vector space of polynomials). The following formula defines an inner product:
$$\langle \vec{p},\vec{q} \rangle = \displaystyle\int_0^{\infty} \vec{p}(x)\vec{q}(x)e^{-x} dx.$$
3. Let $f,g \in C[0,1]$, the functions continuous on the interval $[0,1]$. The following formula defines an inner product:
$$\langle f,g \rangle=\displaystyle\int_0^1 f(x)g(x)dx.$$
4. Let $\{a_k\}_{k=0}^{\infty}, \{b_k\}_{k=0}^{\infty} \in \ell^1(\mathbb{R})$, the set of sequences $\{x_k\}$ such that $\displaystyle\sum_{k=0}^{\infty} |x_k| < \infty$. The following formula defines an inner product:
$$\langle \{a_k\},\{b_k\} \rangle = \displaystyle\sum_{k=0}^{\infty} a_kb_k.$$
Homework 12 (due 18 November):
Problem A. Let $H=(\mathbb{R}^3,\langle \cdot,\cdot \rangle)$ be the inner product space of Example 1. Let $\vec{x}=\begin{bmatrix} 5 \\ 2\\ -1\end{bmatrix}$ and $\vec{y}=\begin{bmatrix} 2 \\ 1 \\ -1\end{bmatrix}$. Compute $\langle \vec{x},\vec{y} \rangle$.
Problem B. Let $H=(\mathbb{P}, \langle \cdot,\cdot \rangle)$ be the inner product space of Example 2. Let $\vec{p}(x)=2x-1$, $\vec{q}(x)=8$. Compute inner product $\langle \vec{p},\vec{q} \rangle$ using integration by parts.
Problem C. Let $H=(\mathscr{C}[0,1],\langle \cdot,\cdot \rangle)$ be the inner product space of Example 3. Let $f(x)=x^2+1$ and $g(x)=x+7$. Calculate $\langle f,g \rangle$.
Problem D. Let $H=(\ell^1(\mathbb{R}),\langle\cdot,\cdot\rangle)$ be the inner product space of Example 4. Let $\{a_k\}_{k=0}^{\infty} = \left\{ \dfrac{1}{3^k} \right\}_{k=0}^{\infty}$ and $\{b_k\}_{k=0}^{\infty} = \left\{ \dfrac{1}{7^k} \right\}_{k=0}^{\infty}$. Calculate $\left\langle \left\{ a_k \right\}, \left\{b_k\right\} \right\rangle$ (hint: this is a geometric series). Let $\left\{c_k\right\}=\left\{d_k\right\}=\sqrt{\dfrac{1}{k!}}$. Calculate $\langle c_k,d_k \rangle$ (hint:recall the power series $e^x = \displaystyle\sum_{k=0}^{\infty} \dfrac{x^k}{k!}$).
Problem E.: Show that set $\left\{ \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 0\\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix} \right\}$ is a mutually orthogonal set of vectors. Also show that the set $\left\{ \begin{bmatrix} 1 \\ 1 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix} \right\}$ is not a mutually orthogonal set of vectors.
Problem F.: Consider the vector space $(\mathbb{P},\langle \cdot,\cdot \rangle)$ where the inner product is given by
$$\langle p(x),q(x) \rangle = \displaystyle\int_{-\infty}^{\infty} p(x)q(x)e^{-x^2} dx.$$
It can be shown (via methods of calculus 2) that the moments in this inner product space are
$$\langle 1,1 \rangle=\sqrt{\pi},$$
$$\langle x,1 \rangle=0,$$
$$\langle x^2,1 \rangle=\dfrac{\sqrt{\pi}}{2},$$
$$\langle x^3,1 \rangle = 0,$$
$$\langle x^4,1 \rangle = \dfrac{3\sqrt{\pi}}{4},$$
$$\langle x^5,1 \rangle = 0,$$
$$\langle x^6,1 \rangle = \dfrac{15\sqrt{\pi}}{8}.$$
Use these moments and the "linear in the first argument" property of inner products to compute $\langle 4x^2+3x+9,1\rangle$ and $\langle 32x^5-64x^3+24x,1\rangle$.
Problem G.: Consider the inner product space $(C[0,1],\langle \cdot,\cdot \rangle)$ where
$$\langle f,g \rangle = \displaystyle\int_0^1 f(x)g(x)x^2 dx.$$
Compute $\mathrm{proj}_{x^2-3x} (5x+2)$ and $\mathrm{proj}_{5x+2}(x^2-3x)$. (hint: using WolframAlpha for integrals is ok here)
