 # Vector Spaces

## 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

## Form

Vectors can be either presented horizontally,

x = (x1, x2, ..., xn)

or vertically.

x = x1 x2 : xn

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.

e1 = (1, 0, ..., 0), e2 = (0, 1, ..., 0), : en = (0, 0, ..., 1)

We also mention the special zero vector

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

## Use

A number (scalar) can also be multiplied to vectors

c (x1, x2, ..., xn) = (cx1, cx2, ..., cxn)

The addition and scalar multiplication can also be combined to form linear combinations c1x1 + c2x2 + ... + ckxk. 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.

e1 = (1, 0, ..., 0), e2 = (0, 1, ..., 0), : en = (0, 0, ..., 1)

We also mention the special zero vector

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

### Example

u = (1, 2), v = (3, 4)

Then

u + v = (1 + 3, 2 + 4) = (4, 6)

### Example

u = (1, 2)

Then

3u = (3*1, 3*2) = (3, 6)

### Example

u = (1, 2), v = (3, 4), w = (5, 6)

Then

u + v + w = (1 + 3 + 5, 2 + 4 + 6) = (9, 12)

### Example

u = (1, 2), v = (3, 4), w = (5, 6)

Then

3u - 4v + w = (3*1 - 4*3 + 5, 3*2 - 4*4 + 6) = (-4, -4)

### Example

x = (1, 0, -3), y = (2, -1, 5)

Then

x + y = (1 + 2, 0 - 1, -3 + 5) = (3, -1, 2)

### Example

x = (1, 0, -3)

Then

-2x = (-2*1, -2*0, -2*(-3)) = (-2, 0, 6)

### Example

x = (1, 0, -3), y = (2, -1, 5)

Then

-2x + 3y = (-2*1 + 3*2, -2*0 + 3*(-1), -2*(-3) + 3*5) = (4, -3, 21)