Description

Inner products are operations from $V \times V \rightarrow \mathbb{R}$. They must fulfill the following 4 properties for any $\vec{x}, \vec{y}, \vec{z} \in V$ and $\alpha \in \mathbb{R}$:

1. Commutativity: $\langle \vec{x}, \vec{y}\rangle = \langle \vec{y}, \vec{x}\rangle$

2. Distributivity: $\langle \vec{x}, \vec{y} + \vec{z}\rangle = \langle \vec{x}, \vec{y}\rangle + \langle \vec{x}, \vec{z}\rangle$

3. Scaling: $\langle \alpha\vec{x}, \vec{y}\rangle = \alpha\langle \vec{x}, \vec{y}\rangle$

4. Nonnegativity: $\langle \vec{x}, \vec{x}\rangle \geq 0$, where $\langle \vec{x}, \vec{x}\rangle = 0$ if and only if $\vec{x} = 0$.

In this class, we are going to use the inner product in the vector space $\mathbb{R}^n$ that is defined as follows (commonly known as the dot product): Let $\vec{x} = \begin{bmatrix}x_1 \\ x_2 \\ \vdots \\ x_n\end{bmatrix}$ and $\vec{y} = \begin{bmatrix}y_1 \\ y_2 \\ \vdots \\ y_n\end{bmatrix}$.

$\langle \vec{x}, \vec{y}\rangle = \left\langle \begin{bmatrix}x_1 \\ x_2 \\ \vdots \\ x_n\end{bmatrix}, \begin{bmatrix}y_1 \\ y_2 \\ \vdots \\ y_n\end{bmatrix}\right\rangle = x_1y_1 + x_2y_2 + \cdots + x_ny_n$