Machine Learning experts cannot live without Linear Algebra:
- ML make heavy use of Scalars
- ML make heavy use of Vectors
- ML make heavy use of Matrices
- ML make heavy use of Tensors
The purpose of this chapter is to highlight the parts of linear algebra that is used in data science projects like machine learning and deep learning.
| Col 1 | Col 2 | Col 3 | Col 4 | Col 5 |
|---|---|---|---|---|
| Scalar | Vector(s) | |||
| 1 | 1 2 3 1 2 3 | |||
| 1 2 3 | 1 2 3 | |||
| 1 | ||||
| 2 | ||||
| 3 | ||||
| 1 | 2 | 3 | ||
| Matrix | Tensor | |||
| 1 2 3 4 5 6 | Tensor 1 2 3 4 5 6 4 5 6 1 2 3 End Table | |||
| 1 | 2 | 3 | ||
| 4 | 5 | 6 | ||
| 1 2 3 4 5 6 | 4 5 6 1 2 3 | |||
| 1 | 2 | 3 | ||
| 4 | 5 | 6 | ||
| 4 | 5 | 6 | ||
| 1 | 2 | 3 |
Vectors and Matrices
Vectors and Matrices are the languages of data.
With ML, most things are done with vectors and matrices.
With vectors and matrices, you can Discover Secrets.
Scalars
In linear algebra, a scalar is a single number.
In JavaScript it can be written like a constant or a variable:
const myScalar = 1;
let x = 1;
var y = 1;
Vectors
In linear algebra, a vector is an array of numbers.
In JavaScript, it can be written as an array:
const myArray = [50,60,70,80,90,100,110,120,130,140,150];
myArray.length; // the length of myArray is 11
An array can have multiple dimensions, but a vector is a 1-dimensional array.
A vector can be written in many ways. The most common are:
| Col 1 | Col 2 | Col 3 |
|---|---|---|
| v = | 1 2 3 | |
| 1 | 2 | 3 |
or:
| Col 1 | Col 2 |
|---|---|
| v = | 1 2 3 |
| 1 | |
| 2 | |
| 3 |
| Col 1 | Col 2 |
|---|---|
| The image to the left is a Vector. The Length shows the Magnitude. The Arrow shows the Direction. |
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Matrices
In linear algebra, a matrix is a 2-dimensional array.
| Col 1 | Col 2 | Col 3 | Col 4 |
|---|---|---|---|
| C = | 3 0 0 0 0 3 0 0 0 0 3 0 0 0 0 3 | ||
| 3 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
In JavaScript, a matrix is typically represented as a two-dimensional (2D) array, meaning an array of arrays. Each inner array represents a row, and each element within those arrays represents a column.
Example
const myArray = [[1,2],[3,4],[5,6]];
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Tensors
A Tensor is an N-dimensional Matrix.
| Col 1 | Col 2 | Col 3 | Col 4 | Col 5 |
|---|---|---|---|---|
| T = | 1 2 3 4 5 6 4 5 6 1 2 3 | |||
| 1 2 3 4 5 6 | 4 5 6 1 2 3 | |||
| 1 | 2 | 3 | ||
| 4 | 5 | 6 | ||
| 4 | 5 | 6 | ||
| 1 | 2 | 3 |
In JavaScript, a tensor is an array with multiple indices (indexes).
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Linear Algebra is the branch of mathematics that concerns linear equations (and linear maps) and their representations in vector spaces and through matrices.
Linear algebra is central to almost all areas of mathematics.
Wikipedia
