# Linear Algebra - Vector Vector Operations

### Table of Contents

## 1 - About

Vector Vector Operations:

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

## 2 - Articles Related

## 3 - 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() |

## 4 - Operations

### 4.1 - Translation

A vector translation is also known as a vector Addition.

#### 4.1.1 - Syntax

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

v + 0 = v

#### 4.1.2 - Property

Vector addition is

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

- and commutative. The order doesn't matter.

x + y = y + x

#### 4.1.3 - Representation

#### 4.1.4 - 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]

### 4.2 - Scalar Multiplication

### 4.3 - Dot product

### 4.4 - 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>

## 5 - Others

### 5.1 - Inner product

## 6 - Cross Property

### 6.1 - 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.