Vectors and Vector Spaces

Algebraic Representations

e.g. $(1,3)$, $\left(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right)$, $(0,0,0)$

Ordered n-tuple: given a positive integer $n$, an ordered sequence of $n$ real (or complex) numbers is called an ordered n-tuple.

Euclidean space: the set of all n-tuples of real numbers is called n-space or Euclidean space, denoted ${\mathbb{R}}^n$.

• the set of all n-tuples of complex numbers is called complex Euclidean space, denoted ${\mathbb{C}}^n$.

A vector in ${\mathbb{R}}^n$ or ${\mathbb{C}}^n$ can be represented as an n-tuple

$v=⟨v_1,v_2,v_3\dots v_n⟩$

where $v_i\in \mathbb{R}$ if $v\in {\mathbb{R}}^n$, or $v_i\in \mathbb{C}$ if $v\in {\mathbb{C}}^n$ for $i=1,2,\dots ,n$. This is known as the algebraic representation of a vector. Each number is called a component or coordinate.

Zero vector $\Rightarrow 0=⟨0,0,0,\dots ,0⟩$

• A vector in ${\mathbb{R}}^n$ is called a real vector
• A vector in ${\mathbb{C}}^n$ is called a complex vector

Geometric Representation

• Real vectors in ${\mathbb{R}}^2$ and ${\mathbb{R}}^3$ can be represented geometrically
• Commonly used in physics where quantities possess magnitude and direction

A vector $v$ is drawn as an arrow from an initial point to a terminal point.

Collinear vectors and parallel vectors.

A zero geometric vector has coincident initial and terminal points. It has a length of zero and no natural direction.