Vectors - Quantum Resources Site

Abstract¶

A very brief overview of vectors, operations of vectors and other key properties.

Keywords:Vectorsdot productcross product.¶

Vectors Definition¶

A vector is a collection of elements from a given field $\mathbb{F}$ . This field could be the real numbers, $\mathbb{R}$ , or the complex number, $\mathbb{C}$ , for example. Vectors are often written in columns such as

\begin{align*} \bm{x} &= \begin{bmatrix} x_{1} \\ x_{2} \\ \vdots \\ x_{n} \end{bmatrix}, \end{align*}

(1)

where each component $x_{i}$ is a element in $\mathbb{F}$ . This vector is an $n$ dimensional vector over the field $\mathbb{F}$ , which we can write as $\bm{x} \in \mathbb{F}^{n}$ , meaning $\bm{x}$ is a vector of $n$ components where each components is from the field $\mathbb{F}$ .

Vector Operations:¶

Vector Addition

Scalar Multiplication

Dot Product

Cross Product

let $\bm{x} \in \mathbb{F}^{n}, ~\bm{y} \in \mathbb{F}^{n}$

\bm{x} + \bm{y} = \begin{bmatrix} x_{1} + y_{1} \\ x_{2} + y_{2} \\ \vdots \\ x_{n} + y_{n} \end{bmatrix}

(2)

Vector Norm

Orthogonality

Span

Let $\bm{x} \in \mathbb{F}^{n}, ~ \lambda \in \mathbb{F}^{1}$

\vert \vert \bm{x} \vert \vert = \bigg( \sum^{n}_{i} x_{i}^{2} \bigg)^{1/2}

(6)

Properties:

$\vert \vert \bm{x} \vert \vert \geq 0, ~\textrm{where}~\vert \vert \bm{x} \vert \vert = 0 ~\textrm{iif.}~\bm{x}=0$
$\vert \vert \lambda \bm{x} \vert \vert = \vert \lambda \vert ~ \vert \vert \bm{x} \vert \vert$
$\vert \vert \bm{x} + \bm{y} \vert \vert \leq \vert \vert \bm{x} \vert \vert + \vert \vert \bm{y} \vert \vert$

This is just one example of a vector norm. See more on the norms page

Vector Properties¶

Pythagoras' Theorem

Cauchy-Schwarz Inequality

let $\bm{x} \in \mathbb{F}^{n}, ~\bm{y} \in \mathbb{F}^{n}$

\vert \vert \bm{x} + \bm{y} \vert \vert^{2} = \vert \vert \bm{x} \vert \vert^{2} + \vert \vert \bm{y} \vert \vert^{2}