# Example Problems

Show that the inner product defined above for $$\mathbb{R}^n$$ as $$\langle \vec{x}, \vec{y}\rangle = x\_1y\_1 + x\_2y\_2 + \cdots + x\_ny\_n$$ fulfills the 4 properties of an inner product.

Let $$\vec{x}, \vec{y}, \vec{z} \in \mathbb{R}^n$$ and $$\alpha \in \mathbb{R}$$.

1. $$\begin{align\*}\langle \vec{y}, \vec{x}\rangle &= y\_1x\_1 + y\_2x\_2 + \cdots + y\_nx\_n \ &= x\_1y\_1 + y\_2x\_2 + \cdots + x\_ny\_n \ &= \langle \vec{x}, \vec{y}\rangle\end{align\*}$$
2. $$\begin{align\*}\langle \vec{x}, \vec{y} + \vec{z}\rangle &= x\_1(y\_1 + z\_1) + x\_2(y\_2 + z\_2) + \cdots + x\_n(y\_n + z\_n) \ &= x\_1y\_1 + x\_2y\_2 + \cdots + x\_ny\_n + x\_1z\_1 + x\_2z\_2 + \cdots + x\_nz\_n \ &= \langle \vec{x}, \vec{y}\rangle + \langle \vec{x}, \vec{z}\rangle\end{align\*}$$
3. $$\begin{align\*}\langle \alpha\vec{x}, \vec{y}\rangle &= \alpha x\_1y\_1 + \alpha x\_2y\_2 + \cdots + \alpha x\_ny\_n \ &= \alpha(x\_1y\_1 + x\_2y\_2 + \cdots + x\_ny\_n) \ &= \alpha \langle \vec{x}, \vec{y}\rangle\end{align\*}$$
4. $$\langle \vec{x}, \vec{x}\rangle = x\_1^2 + x\_2^2 + \cdots x\_n^2$$.

   Since every component is squared, we know that $$\langle \vec{x}, \vec{x}\rangle \geq 0$$. Let us check when $$x\_1^2 + x\_2^2 + \cdots x\_n^2 = 0$$. Since we are adding each component squared, all components must be $$0$$ in order for this sum to be $$0$$ as well, i.e. $$\vec{x}$$ must be the zero vector. Therefore, $$\langle \vec{x}, \vec{x}\rangle = 0 \iff \vec{x} = \vec{0}$$.

## Norm

The norm of a vector $$\vec{x} \in V$$ is its length. We denote the norm as $$|\vec{x}|: V \rightarrow \mathbb{R}$$.

For vectors in $$\mathbb{R}^n$$, to find the length of a vector $$\vec{x} = \begin{bmatrix}x\_1 \ x\_2 \ \vdots \ x\_n\end{bmatrix}$$, we use the Pythagorean theorem.

$$|\vec{x}| = \sqrt{x\_1^2 + x\_2^2 + \cdots + x\_n^2} = \sqrt{\langle \vec{x}, \vec{x}\rangle}$$

Therefore, we can express the inner product of $$\vec{x}$$ with itself as the square of its norm:

$$\langle \vec{x}, \vec{x}\rangle = |\vec{x}|^2$$

### Properties

Norms have to fulfill 3 properties for $$\vec{x}, \vec{y} \in V$$ and $$\alpha \in \mathbb{R}$$:

1. Nonnegativity: $$|\vec{x}| \geq 0$$, where $$|\vec{x}| = 0$$ if and only if $$\vec{x} = \vec{0}$$
2. Scaling: $$|\alpha \vec{x}| = |\alpha||\vec{x}|$$
3. Triangle inequality: $$|\vec{x} + \vec{y}| \leq |\vec{x}| + |\vec{y}|$$, where $$|\vec{x} + \vec{y}| = |\vec{x}| + |\vec{y}|$$ if and only if $$\vec{x} = \alpha\vec{y}$$ (i.e. when $$\vec{x}$$ and $$\vec{y}$$ are parallel)

## Angle

We define the angle $$\theta$$ between 2 vectors $$\vec{x}, \vec{y}$$ using the following equation:

$$\langle \vec{x}, \vec{y}\rangle = |\vec{x}||\vec{y}|\cos\theta$$

$$\cos\theta = \frac{\langle \vec{x}, \vec{y}\rangle}{|\vec{x}||\vec{y}|}$$

![Angle between 2 vectors](https://upload.wikimedia.org/wikipedia/commons/2/25/Scalar-product-dot-product.svg)


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