## Introduction

A vector space represents the idea of a set of vectors, closed under certain ways of combining them. A vector space is a set V, together with a binary operation (+) from V x V -> V and a binary operation F x V -> V where F is a field, such that the following axioms hold.

- ∀x, y ∈ V, x + y ∈ V
- ∀x, y, z ∈ V, x + (y + z) = (x + y) + z
- ∃0 ∈ V, ∀x ∈ V, x + 0 = x
- ∀x ∈ V, ∃-x ∈ V, x + -x = 0
- ∀a ∈ F, ∃x ∈ V, a * x ∈ V

## Contents

## Form

Vectors can be either presented horizontally,

_{1}, x

_{2}, ..., x

_{n})

or vertically.

_{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.

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

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

_{n}= (0, 0, ..., 1)

We also mention the special zero vector

## Use

A number (scalar) can also be multiplied to vectors

_{1}, x

_{2}, ..., x

_{n}) = (cx

_{1}, cx

_{2}, ..., cx

_{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}. 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.

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

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

_{n}= (0, 0, ..., 1)

We also mention the special zero vector

### Example

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### Example

Then