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

We concluded our previous discussion about inconsistent linear systems by saying that no solution exists for such a system. However, there are scenarios in which it’s still important to find the solution for a system, even if it’s an approximate solution. Imagine a space travel company is landing its rocket booster back on earth. The company’s goal is to land the rocket on a platform floating on the sea. Assume that the mathematical model for this task is a linear system with variables like wind speed, rocket weight, and sea waves. The ideal case would be to land the rocket on the marker at the center of the platform.

However, let’s assume that the linear system turns out to be an inconsistent system. Now, instead of aborting the landing because an exact solution doesn’t exist to land the rocket on the marked point, it’s in the company’s best interest to land the rocket anywhere on the platform. This may be achieved by calculating an approximate solution for the linear system.


## Approximation
In contrast to the exact solution that satisfies all equations of the linear system, an approximate solution is a point that nearly satisfies all the equations.


A point satisfies an equation if, by replacing the variables of the equation with the values of the point, the equality is maintained.





| **Exact Solution** | **Near Satisfaction**  |
| :- | :- |
| $a_1x_1+a_2x_2+\cdots+a_nx_n = b$ $a_1x_1+a_2x_2+\cdots+a_nx_n -b = 0$|$a_1\hat x_1+a_2\hat x_2+\cdots+a_n\hat x_n \approx b$ $a_1\hat x_1+a_2\hat x_2+\cdots+a_n\hat x_n -b = \beta$ |
| $(x_1,x_2,\cdots,x_n)$ is the exact solution | $(\hat x_1,\hat x_2,\cdots,\hat x_n)$ is the approximate solution, while $\lvert{\beta}\rvert$ is as small as possible.|




The value of $|\beta|$ can be explained as the **algebraic distance** between the solution point and the equation. In the case of exact solutions, $\beta$ has the ideal value of $0$. Geometrically, the solution point is incident to the linear equation. In contrast, for approximation, the point resulting in the smallest value of $\beta$ nearly satisfies the equation.


# Approximation
In contrast to the exact solution that satisfies all equations of the linear system, an approximate solution is a point that nearly satisfies all the equations.


A point satisfies an equation if, by replacing the variables of the equation with the values of the point, the equality is maintained.





| **Exact Solution** | **Near Satisfaction**  |
| :- | :- |
| $a_1x_1+a_2x_2+\cdots+a_nx_n = b$ $a_1x_1+a_2x_2+\cdots+a_nx_n -b = 0$|$a_1\hat x_1+a_2\hat x_2+\cdots+a_n\hat x_n \approx b$ $a_1\hat x_1+a_2\hat x_2+\cdots+a_n\hat x_n -b = \beta$ |
| $(x_1,x_2,\cdots,x_n)$ is the exact solution | $(\hat x_1,\hat x_2,\cdots,\hat x_n)$ is the approximate solution, while $\lvert{\beta}\rvert$ is as small as possible.|




The value of $|\beta|$ can be explained as the **algebraic distance** between the solution point and the equation. In the case of exact solutions, $\beta$ has the ideal value of $0$. Geometrically, the solution point is incident to the linear equation. In contrast, for approximation, the point resulting in the smallest value of $\beta$ nearly satisfies the equation.

Explore the idea of an approximate solution for an inconsistent linear system.

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

Approximate Solution

Approximation