Scalar Product

A scalar product is a special type of operation that acts on two vectors, and returns a scalar result. Scalar products are denoted as an ordered pair between angle-brackets: <x,y>. A scalar product between vectors must satisfy the following four rules:

1. ${\displaystyle \langle x,x\rangle \geq 0,\quad \forall x\in V}$
2. ${\displaystyle \langle x,x\rangle =0}$, only if x = 0
3. ${\displaystyle \langle x,y\rangle =\langle y,x\rangle }$
4. ${\displaystyle \langle x,cy_{1}+dy_{2}\rangle =c\langle x,y_{1}\rangle +d\langle x,y_{2}\rangle }$

If an operation satisifes all these requirements, then it is a scalar product.

Norm

The norm is an important scalar quantity that indicates the magnitude of the vector. Norms of a vector are typically denoted as ${\displaystyle \|x\|}$. To be a norm, an operation must satisfy the following four conditions:

1. ${\displaystyle \|x\|\geq 0}$
2. ${\displaystyle \|x\|=0}$ only if x = 0.
3. ${\displaystyle \|cx\|=|c|\|x\|}$
4. ${\displaystyle \|x+y\|\leq \|x\|+\|y\|}$

A vector is called normal if it's norm is 1. A normal vector is sometimes also referred to as a unit vector. Both notations will be used in this book. To make a vector normal, but keep it pointing in the same direction, we can divide the vector by its norm:

${\displaystyle {\bar {x}}={\frac {x}{\|x\|}}}$

Orthogonality

Two vectors x and y are said to be orthogonal if the scalar product of the two is equal to zero:

${\displaystyle \langle x,y\rangle =0}$

Two vectors are said to be orthonormal if their scalar product is zero, and both vectors are unit vectors.

Cauchy-Schwartz Inequality

The cauchy-schwartz inequality is an important result, and relates the norm of a vector to the scalar product:

${\displaystyle |\langle x,y\rangle |\leq \|x\|\|y\|}$

Metric (Distance)

The distance between two vectors in the vector space V, called the metric of the two vectors, is denoted by d(x, y). A metric operation must satisfy the following four conditions:

1. ${\displaystyle d(x,y)\geq 0}$
2. ${\displaystyle d(x,y)=0}$ only if x = y
3. ${\displaystyle d(x,y)=d(y,x)}$
4. ${\displaystyle d(x,y)\leq d(x,z)+d(z,y)}$