The `n`-dimensional (real) euclidean space **R**^{n}
is the collection of `n`-tuples of numbers, called vectors. Vectors
can be either presented horizontally,

` x` = (

or vertically.

= [x |
x_{1} |
] |

x_{2} |
||

: | ||

x_{n} |

The only thing you need to keep in mind is that, when vectors are mixed with matrices, it is often more convenient to use vertical vectors.

The number `n` is the dimension of the vector ` x`.
The numbers

**e**_{1} = (1, 0, ..., 0),

**e**_{2} = (0, 1, ..., 0),

:

` e_{n}` = (0, 0, ..., 1).

We also mention the special zero vector

**0** = (0, 0, ..., 0).

Example The 1-dimensional euclidean space

**R**^{1} = {(`x`): `x` is a real number}

can be identified with all real numbers **R** and be visualized as a straight line.

The 2-dimensional euclidean space

**R**^{2} = {(`x`_{1}, `x`_{2}):
`x`_{1}, `x`_{2} are real numbers}

can be visualized as the coordinate plane.

The 3-dimensional euclidean space

**R**^{3} = {(`x`_{1}, `x`_{2},
`x`_{3}): `x`_{1}, `x`_{2}, `x`_{3}
are real numbers}

can be visualized as the world we are living in.

The 0-dimensional euclidean space **R**^{0} consists of a single point - the origin.

The example illustrates that the euclidean vectors can be geometrically presented (in a coordinate system) as an arrow starting from the origin. If an arrow does not start from the origin, we need to parallel shift it so that it begins at the origin.

Vectors of the same dimension can be added

(`x`_{1}, `x`_{2}, ..., `x _{n}`) +
(

A number (scalar) can also be multiplied to vectors

`c` (`x`_{1}, `x`_{2}, ..., `x _{n}`)
= (

The addition and scalar multiplication can also be combined to form linear combinations
`c`_{1}**x**_{1} + `c`_{2}**x**_{2}
+ ... + `c _{k}x_{k}`.

Example
Let ` u` = (1, 2),

` u` +

3

3

` x` +

-2

-2

Moreover, because the dimension of ` u`,

Geometrically, the addition is described by parallelogram, and the scalar multiplication is the stretching/shrinking and sometimes reversing (for negative scalars) of vectors.

This picture shows the linear combinations of two (nonparallel) vectors. It suggests that all the linear combinations of the two vectors form a plane passing through the origin.

Finally, the operations have the following obvious properties.

- commutative
+**u**=**v**+**v****u**- associative
- (
+**u**) +**v**=**w**+ (**u**+**v**)**w** - zero
+**u****0**==**u****0**+**u**- distributive
`r`(+**u**) =**v**`r`+**u**`r`**v**- distributive
- (
`r`+`s`)=**u**`r`+**u**`s`**u** - associative
- (
`rs`)=**u**`r`(`s`)**u**

[Extra Vector addition is given by parallelogram]