Gain insights into linear algebra essentials for data science, focusing on vectors, matrices, and tensors. Explore practical Python applications, engaging visuals, real datasets, and hands-on projects.

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Linear algebra is a fundamental pillar of data science. In advanced models in data science, like neural networks, the inputs and transformations are based upon vectors, matrices, and tensors which require a reasonable understanding of linear algebra to get the desired results. It is elegant and the most applied mathematics under the umbrella of data science.

This course teaches linear algebra with a focus on data science. This course encompasses several engaging illustrations, including static images and animations. Furthermore, this course presents mathematical modeling through programming in Python. This course contains several executable coding playgrounds on real data sets and a final project with practical applications.

Aside from theoretical implementations, the modern-day world needs its daunting calculations, trajectory mapping, and distance manipulation, all of which linear algebra provides. By the end of this course, you’ll have a working knowledge of all the necessary teachings in linear algebra.

Linear Algebra for Data Science Using Python

## Modulo operation
Given $a, m, r \in \Z$, $m > 0$, and $0 \le r \le m-1$, we write

$$a\ \text{mod}\ m \equiv r$$ 

when $m$ divides $a-r$ (_also denoted as_ $m | (a-r)$). Here, $r$ is referred to as the remainder.
### Example 
Given $a=17, m=5$, find r.

One possible answer is  $r=2$ because 
$17 \mod 5 \equiv 2$. This is because 5|(17-2), which is read as 5 divided by 15.  

For the example above, we can find many other remainders, such as 7 and -3. This is because 5|(17-7) and 5|(17+3).

 
## Finite field
A **finite field** is a field with a finite number of elements. Finite fields have many applications in data science and especially in cryptography. We usually define a finite field using modulo operation.

### Galois field
A finite field having prime number of elements over mod $p$ is called a **Galois field** or **prime field**. A Galois field is usually referred to as $GF(p)$, where $p$ is a prime number and $GF(p)=\{0, 1, \cdots, p-1\}$.

### Example 
A set $S=\{0,1,2,3,4\}$ defined over mod  $5$ makes a Galois field $GF(5)$. This implies that after each operation on field, we apply mod $5$ on the result. For instance, 

$$2+4 \equiv 6 \mod 5 \equiv 1$$ 

and

$$2\times 4 \equiv 8 \mod 5 \equiv 3$$

Let’s now provide verification of each property using Python code.

#### Axiom verification
We can prove the axiom verification by using mathematical arguments, but let’s verify it through code and test all the properties of the field $S$.
##### Closure
We define the `sum` function, which computes the sum of two elements of a field and then computes `mod 5` of the result. We first make a list of the elements, `S=[0,1,2,3,4]`, and then use `itertools` to generate all possible pairs. The following code calls the `sum` function on every possible pair of elements of S, verifying the closure axiom.
Similarly, we can test the closure under multiplication using `mul`, which also first multiplies two numbers and then takes their modulus.

Below is the complete code for testing closure under $+$ and $\times$.

# Modulo operation
Given $a, m, r \in \Z$, $m > 0$, and $0 \le r \le m-1$, we write

$$a\ \text{mod}\ m \equiv r$$ 

when $m$ divides $a-r$ (_also denoted as_ $m | (a-r)$). Here, $r$ is referred to as the remainder.
## Example 
Given $a=17, m=5$, find r.

One possible answer is  $r=2$ because 
$17 \mod 5 \equiv 2$. This is because 5|(17-2), which is read as 5 divided by 15.  

For the example above, we can find many other remainders, such as 7 and -3. This is because 5|(17-7) and 5|(17+3).

 
# Finite field
A **finite field** is a field with a finite number of elements. Finite fields have many applications in data science and especially in cryptography. We usually define a finite field using modulo operation.

## Galois field
A finite field having prime number of elements over mod $p$ is called a **Galois field** or **prime field**. A Galois field is usually referred to as $GF(p)$, where $p$ is a prime number and $GF(p)=\{0, 1, \cdots, p-1\}$.

## Example 
A set $S=\{0,1,2,3,4\}$ defined over mod  $5$ makes a Galois field $GF(5)$. This implies that after each operation on field, we apply mod $5$ on the result. For instance, 

$$2+4 \equiv 6 \mod 5 \equiv 1$$ 

and

$$2\times 4 \equiv 8 \mod 5 \equiv 3$$

Let’s now provide verification of each property using Python code.

### Axiom verification
We can prove the axiom verification by using mathematical arguments, but let’s verify it through code and test all the properties of the field $S$.
#### Closure
We define the `sum` function, which computes the sum of two elements of a field and then computes `mod 5` of the result. We first make a list of the elements, `S=[0,1,2,3,4]`, and then use `itertools` to generate all possible pairs. The following code calls the `sum` function on every possible pair of elements of S, verifying the closure axiom.
Similarly, we can test the closure under multiplication using `mul`, which also first multiplies two numbers and then takes their modulus.

Below is the complete code for testing closure under $+$ and $\times$.

Learn about modular arithmetic and finite fields with examples.

Introduction

Linearity

Matrices

Solving Linear Systems

Singularity

Linear Regression and Least Squares

Vector Space

Vector Spaces of a Matrix

Singular Value Decomposition: SVD

Learning to Find Discriminative Null Space for Face Recognition

Finite Field

Modulo operation