# Vector space

A **vector space (linear space)** is a set whose elements are called vectors.

Two operations are defined in a vector space

**Vector addition**

In a vector space the vectors may be added together**Scalar multiplication**

In a vector space the vectors may be multiplied by numbers called scalars (e.g. integers, real numbers, complex numbers, etc).

The **dimension of a vector space** is the number (n) of independent directions in the space.

**Example**. In a vector space of two dimensions (n=2) the vectors move in two independent directions (x, y axes of the plane).

In a three-dimensional vector space (n=3), vectors move in three independent directions (x, y, z axes of space).

A vector space can be finite or infinite dimensional

**finite-dimensional**

if dimension is a natural number n∈N**infinite-dimensional**

if dimension is an infinite cardinale n=∞

Each element of the n-dimensional vector space is a vector **v** = (x_{1},x_{2},...,x_{n}) that is a tuple of n real number (vector), where x_{1},x_{2},...,x_{n} are called coordinates, elements or components of vector **v**.

The set of all n-tuples of real numbers **v** = (x_{1},x_{2},...,x_{n}) is called **n-space** and it's denoted by R^{n}.

**Examples**. The vector v=(1,2) belong to R^{2}. The vector v=(3,2,1) belong to R^{3}

In a vector space **two vectors are equal** if the corresponding components are equal.

**Examples**. The following vectors u=(3,2,1) and v=(3,2,1) are equal. The following vectors u=(3,2,1) and are v=(2,3,1) not equal, because x_{1u}=3≠x_{1v}=2 and x_{2u}=2≠x_{2v}=3

A particular vector is the **zero vector**

The zero vector is a vector (0,0, ... 0) in which all components are null.