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

# Orthogonal subspaces
Two subspaces, $U$ and $V$, of a vector space are *orthogonal* when every vector in $U$ is orthogonal to every vector in $V$ and the other way around.

## Example
The $x$-axis and $y$-axis in $\R^2$ are orthogonal subspaces. In particular, let $\bold v=\begin{bmatrix}\alpha\\0\end{bmatrix}$ and $\bold u=\begin{bmatrix}0\\\beta\end{bmatrix}$ be any vectors along the $x$-axis and $y$-axis, respectively. We can see that the dot product of $\bold{u}$ and $\bold{v}$ is $0$, implying that $\bold{u}$ and $\bold{v}$ are orthogonal.

>**Note:** It's sufficient to test the orthogonality of the basis of two subspaces to establish the orthogonality of the subspaces.

# Planes at the right angle
We may make a mistake if we visualize two planes in $\R^3$ that look orthogonal, as shown in the visualization below. However, these subspaces aren't orthogonal, because a line of intersection is common to both the planes. The vectors along the line of intersection break the orthogonality between the subspaces. 

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

Orthogonal Spaces and Complements

Orthogonal subspaces

Example