# Linear Algebra - Vector Vector Operations

### Table of Contents

## About

Vector Vector Operations:

- Translation (also known as addition or substraction)
- Scalar Multiplication (Scaling)

## Articles Related

## List

Using operator overloading, you can compute this operations with the following syntax.

Operation | Syntax |
---|---|

vector addition | u+v |

vector negation | -v |

vector subtraction | u-v |

scalar-vector multiplication | alpha*v |

division of a vector by a scalar | v/alpha |

dot-product | u*v |

getting value of an entry | v[d] |

setting value of an entry | v[d] = … |

testing vector equality | u == v |

pretty-printing a vector | print(v) |

copying a vector | v.copy() |

## Operations

### Translation

A vector translation is also known as a vector Addition.

#### Syntax

`[u1, u2, . . . , un] + [v1, v2, . . . , vn] = [u1 + v1, u2 + v2, . . . , un + vn]`

`v + 0 = v`

#### Property

Vector addition is

`(x + y) + z = x + (y + z)`

- and (Function|Operator) - Algebraic (Laws|properties) - Axioms. The order doesn't matter.

`x + y = y + x`

#### Representation

#### Computation

In python:

- For two n-vectors

`def addn(v, w): return [v[i]+w[i] for i in range(len(v))]`

- For n n-vectors:

```
>>> vectorList = [[1,2,3],[1,2,3], [1,2,3]]
# How to group the number by position in the vector
>>> [[i[j] for i in vectorList] for j in range(len(vectorList[0])) ]
[[1, 1, 1], [2, 2, 2], [3, 3, 3]]
# Then add a sum to the above statement
>>> [sum([i[j] for i in vectorList]) for j in range(len(vectorList[0])) ]
[3, 6, 9]
```

### Scalar Multiplication

### Dot product

### Element-wise multiplication

Element-wise multiplication is the default method when two NumPy arrays are multiplied together.

The element-wise calculation is as follows: <MATH> \mathbf{x} \odot \mathbf{y} = \begin{bmatrix} x_1 . y_1 \\\ x_2 . y_2 \\\ \vdots \\\ x_n . y_n \end{bmatrix} </MATH>

Example: <MATH> \begin{bmatrix} 1 \\\ 2 \\\ 3 \end{bmatrix} \odot \begin{bmatrix} 4 \\\ 5 \\\ 6 \end{bmatrix} = \begin{bmatrix} 4 \\\ 10 \\\ 18 \end{bmatrix} </MATH>

## Others

### Inner product

## Cross Property

### Distributivity

Scalar-vector multiplication distributes over vector addition (translation):

<MATH>\alpha(u+v)=\alpha.u+\alpha.v</MATH>

`2([1, 2, 3] + [3, 4, 4]) = 2 [1, 2, 3] + 2 [3, 4, 4] = [2, 4, 6] + [6, 8, 8] = [8, 12, 14]`

Addition and scalar multiplication are used to defined a a line that not necessarily go through the origin.