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.

linearalgebra.tar.gz

Code

visualizing-planes-3D

line-graph

convex-combinations-animation

intersecting planes

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

## Definition

The **inverse** of a square matrix $A$ is another matrix, $A^{-1}$ of the same size that converts $A$ to identity, $I$, upon multiplication. Mathematically, 

$$A^{-1}A = AA^{-1}= I$$

The matrix, $A^{-1}$, has an opposite effect to that of $A$. Whatever $A$ does, $A^{-1}$ undoes.

The Python code to calculate the inverse of a square matrix is as follows:

import numpy as np

mat = np.array([[1, 0, 0],
                [0, 1, 0],
                [0, 0, 2]])
print(mat)
inv = np.linalg.inv(mat)    
print(f"""The inverse of the matrix is\n{inv}""")

## Example

For example, the elimination matrix $E$ represents two operations, $\bold{r}_3 = \bold{r}_3 + 2\bold{r}_2$ and $\bold{r}_1 = 3\bold{r}_1$: 
$$E=\begin{bmatrix} 3 & 0 & 0\\ 0 & 1 & 0\\0 & 2 & 1 \end{bmatrix}$$

The inverse of this matrix, $E^{-1}$ , is a matrix that does the reverse operations, $\bold{r}_3 = \bold{r}_3 - 2\bold{r}_2$ and $\bold{r}_1 = \frac{1}{3}\bold{r}_1$:
$$E^{-1}=\begin{bmatrix} \frac{1}{3} & 0 & 0\\ 0 & 1 & 0\\0 & -2 & 1 \end{bmatrix}$$

We can verify whether we have the correct inverse by checking if the following equation is true:






$$ EE^{-1}= \begin{bmatrix} 3 & 0 & 0\\ 0 & 1 & 0\\0 & 2 & 1 \end{bmatrix} \begin{bmatrix} \frac{1}{3} & 0 & 0\\ 0 & 1 & 0\\0 & -2 & 1 \end{bmatrix}= \begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&1\end{bmatrix}= I$$


## Uses of inverse

There are numerous uses of the inverse of a matrix. We’ll discuss two here.

### Regeneration 
The inverse of a matrix reverses the effect of the original matrix. For example, given an augmented matrix,  $A_{n \times m}$, if we left multiply it with the matrix $B_{n \times n}$, we transform $A_{n \times m}$ to $A^{'}_{n \times m}$. Mathematically,

$$B_{n \times n}A_{n \times m}=A^{'}_{n \times m} $$ 

In the case that B is invertible, we can regenerate the original matrix by left multiplying the transformed matrix with the inverse of B. Mathematically,

$$B^{-1}_{n \times n}A^{'}_{n \times m}= A_{n \times m}$$


### Solving systems of linear equations

Another use of inverse matrices is related to the solution set of a linear system. Multiply the equation $Ax=b$ by $A^{-1}$ on both sides. 
$$Ax=b \implies A^{-1}Ax=A^{-1}b\implies x = A^{-1}b$$
The solution set of a linear system can be calculated by $A^{-1}b$, given that $A^{-1}$ exists.




# Definition

The **inverse** of a square matrix $A$ is another matrix, $A^{-1}$ of the same size that converts $A$ to identity, $I$, upon multiplication. Mathematically, 

$$A^{-1}A = AA^{-1}= I$$

The matrix, $A^{-1}$, has an opposite effect to that of $A$. Whatever $A$ does, $A^{-1}$ undoes.

The Python code to calculate the inverse of a square matrix is as follows:

# Example

For example, the elimination matrix $E$ represents two operations, $\bold{r}_3 = \bold{r}_3 + 2\bold{r}_2$ and $\bold{r}_1 = 3\bold{r}_1$: 
$$E=\begin{bmatrix} 3 & 0 & 0\\ 0 & 1 & 0\\0 & 2 & 1 \end{bmatrix}$$

The inverse of this matrix, $E^{-1}$ , is a matrix that does the reverse operations, $\bold{r}_3 = \bold{r}_3 - 2\bold{r}_2$ and $\bold{r}_1 = \frac{1}{3}\bold{r}_1$:
$$E^{-1}=\begin{bmatrix} \frac{1}{3} & 0 & 0\\ 0 & 1 & 0\\0 & -2 & 1 \end{bmatrix}$$

We can verify whether we have the correct inverse by checking if the following equation is true:






$$ EE^{-1}= \begin{bmatrix} 3 & 0 & 0\\ 0 & 1 & 0\\0 & 2 & 1 \end{bmatrix} \begin{bmatrix} \frac{1}{3} & 0 & 0\\ 0 & 1 & 0\\0 & -2 & 1 \end{bmatrix}= \begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&1\end{bmatrix}= I$$


# Uses of inverse

There are numerous uses of the inverse of a matrix. We’ll discuss two here.

## Regeneration 
The inverse of a matrix reverses the effect of the original matrix. For example, given an augmented matrix,  $A_{n \times m}$, if we left multiply it with the matrix $B_{n \times n}$, we transform $A_{n \times m}$ to $A^{'}_{n \times m}$. Mathematically,

$$B_{n \times n}A_{n \times m}=A^{'}_{n \times m} $$ 

In the case that B is invertible, we can regenerate the original matrix by left multiplying the transformed matrix with the inverse of B. Mathematically,

$$B^{-1}_{n \times n}A^{'}_{n \times m}= A_{n \times m}$$


## Solving systems of linear equations

Another use of inverse matrices is related to the solution set of a linear system. Multiply the equation $Ax=b$ by $A^{-1}$ on both sides. 
$$Ax=b \implies A^{-1}Ax=A^{-1}b\implies x = A^{-1}b$$
The solution set of a linear system can be calculated by $A^{-1}b$, given that $A^{-1}$ exists.




Learn about invertibility, the properties of inverse matrices, and how to create an algorithm to generate the inverse of a matrix.

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

Inverse of a Square Matrix

Definition

Example