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

## Group 
A group G, over a binary operation, $o$, and two elements $a, b \in G$, is defined with the following four axioms:



* **Closure**: A group is closed under $o$, that is, the result of the operation is also a member of that group.
$$ a\ o\ b = c \in G$$

* **Associativity**: The result of a binary operation on three or more elements remains the same, regardless of the arrangement of parentheses.
$$ (a\ o\ b)\ o\ c = a\ o\ (b\ o\ c), \forall\ c \in G $$ 


* **Identity**: There exists an identity element, $i$, under operation, $o$ such that operation between any element of the group, $c$, and $i$ results in $c$. 
$$\exists\ i \in G\ |\ c\ o\ i = c, \forall\ c \in G  $$



* **Inverse**: For each element, $c$, of the group, there exists an inverse element $\bar c$ such that  operation between $c$ and $\bar c$ results in identity element, $i$.

$$ \forall\ c \in G, \exists\ \bar c\ |\ c\ o\ \bar c = i$$

> **Note:** In general, a scalar can be considered an element of a group and isn’t restricted to a real or complex number.

## Abelian group
An abelian group is a group, $A$, under binary operation, $o$, with one additional axiom.

* **Commutativity**: The order of elements while applying a binary operation is irrelevant.
$$a\ o\ b = b\ o\ a, \forall\ a, b \in A$$


## Field
A field, $F$, satisfies the following axioms:
* **Additive group:** All elements of $F$ form an abelian group under operation $+$, with $0$ as the additive identity.
* **Multiplicative group:** All elements of $F$, excluding $0$, form an abelian group under operation $\times$, with $1$ as the multiplicative identity.
* **Distribution of $\times$ over $+$:** 
Multiplying an element $a$ with the result of the addition of two elements $b$ and $c$ gives the same result as multiplying each element with $a$ first and then adding the results.

$$a\times (b+c)=(a\times b)+(a\times c), \forall\ a,b,c \in F $$

## Examples
Below are some examples of fields.
### Real numbers 
The set of real numbers, $\R$, is a field with standard definitions of $+$ and $\times$.
* $\R$ is closed under $+$ and $\times$ because we're adding or multiplying two real numbers results in a real number.
* Associativity holds in $\R$ for both $+$ and $\times$.
* Commutativity holds in $\R$ for both $+$ and $\times$.
* The numbers $0$ and $1$ are additive and multiplicative identities, respectively.
* The additive inverse of every element is the sign-flipped version of it, whereas the multiplicative inverse is its reciprocal.
* Distributivity of $\times$ over $+$ holds in $\R$.

### Complex numbers
The set of complex numbers, $C$, and the set of rational numbers, $Q$, are fields with standard definitions of $+$ and $\times$.

### Orthogonal matrices
A set of orthogonal matrices of given dimensions are fields. Here, the zero matrix is the additive identity, and the identity matrix is  the multiplicative identity. Additive inverse is found by simply changing the signs of each entry. Finally, by definition, orthogonal matrices are always invertible. 







# Group 
A group G, over a binary operation, $o$, and two elements $a, b \in G$, is defined with the following four axioms:



* **Closure**: A group is closed under $o$, that is, the result of the operation is also a member of that group.
$$ a\ o\ b = c \in G$$

* **Associativity**: The result of a binary operation on three or more elements remains the same, regardless of the arrangement of parentheses.
$$ (a\ o\ b)\ o\ c = a\ o\ (b\ o\ c), \forall\ c \in G $$ 


* **Identity**: There exists an identity element, $i$, under operation, $o$ such that operation between any element of the group, $c$, and $i$ results in $c$. 
$$\exists\ i \in G\ |\ c\ o\ i = c, \forall\ c \in G  $$



* **Inverse**: For each element, $c$, of the group, there exists an inverse element $\bar c$ such that  operation between $c$ and $\bar c$ results in identity element, $i$.

$$ \forall\ c \in G, \exists\ \bar c\ |\ c\ o\ \bar c = i$$

> **Note:** In general, a scalar can be considered an element of a group and isn’t restricted to a real or complex number.

# Abelian group
An abelian group is a group, $A$, under binary operation, $o$, with one additional axiom.

* **Commutativity**: The order of elements while applying a binary operation is irrelevant.
$$a\ o\ b = b\ o\ a, \forall\ a, b \in A$$


# Field
A field, $F$, satisfies the following axioms:
* **Additive group:** All elements of $F$ form an abelian group under operation $+$, with $0$ as the additive identity.
* **Multiplicative group:** All elements of $F$, excluding $0$, form an abelian group under operation $\times$, with $1$ as the multiplicative identity.
* **Distribution of $\times$ over $+$:** 
Multiplying an element $a$ with the result of the addition of two elements $b$ and $c$ gives the same result as multiplying each element with $a$ first and then adding the results.

$$a\times (b+c)=(a\times b)+(a\times c), \forall\ a,b,c \in F $$

# Examples
Below are some examples of fields.
## Real numbers 
The set of real numbers, $\R$, is a field with standard definitions of $+$ and $\times$.
* $\R$ is closed under $+$ and $\times$ because we're adding or multiplying two real numbers results in a real number.
* Associativity holds in $\R$ for both $+$ and $\times$.
* Commutativity holds in $\R$ for both $+$ and $\times$.
* The numbers $0$ and $1$ are additive and multiplicative identities, respectively.
* The additive inverse of every element is the sign-flipped version of it, whereas the multiplicative inverse is its reciprocal.
* Distributivity of $\times$ over $+$ holds in $\R$.

## Complex numbers
The set of complex numbers, $C$, and the set of rational numbers, $Q$, are fields with standard definitions of $+$ and $\times$.

## Orthogonal matrices
A set of orthogonal matrices of given dimensions are fields. Here, the zero matrix is the additive identity, and the identity matrix is  the multiplicative identity. Additive inverse is found by simply changing the signs of each entry. Finally, by definition, orthogonal matrices are always invertible. 







Explore the abstract algebra objects group, abelian group, and field 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

Group and Field

Group