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# 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 $$\vec{y} = \begin{bmatrix}y\_1 \ y\_2 \ \vdots \ y\_n\end{bmatrix}$$.

$$\langle \vec{x}, \vec{y}\rangle = \left\langle \begin{bmatrix}x\_1 \ x\_2 \ \vdots \ x\_n\end{bmatrix}, \begin{bmatrix}y\_1 \ y\_2 \ \vdots \ y\_n\end{bmatrix}\right\rangle = x\_1y\_1 + x\_2y\_2 + \cdots + x\_ny\_n$$