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

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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

# Subspace
A **subspace** is a subset of a vector space, while it also satisfies all the axioms of a vector space. Thus, a subspace is a vector space by itself.

>**Note:** For subspaces of $\R^d$, it’s sufficient only to show the closure under addition and scalar multiplication, or closure under linear combinations.

>**Note:** Every vector space, and therefore every subspace, must have a _zero vector_. In the case of  $\R^d$, this is the *origin*.

## Examples
Every geometric structure, such as lines, planes, circles, and so on, are sets of points. Some of these structures that contain the origin are subspaces.

### Line through the origin
A line through the origin in $\R^d$ is a subspace of $\R^d$. A line in $\R^d$ can be represented by a reference point, $\bold{p}=\begin{bmatrix}x_{1} & x_{2}&\cdots&x_{d}\end{bmatrix}^T$, and a vector in the direction of the line, $\bold{v}=\begin{bmatrix}v_{1} & v_{2}&\cdots&v_{d}\end{bmatrix}^T$ . For example, the following is a subspace:

$$L=\{\bold{p+\alpha v}|\alpha \in \R\}$$

If the line, $L$, passes through the origin, then the reference point, $\bold{p}$, can be set to the zero vector. Then, $L$ can be seen as a span of $\bold{v}$. We’ve already discussed that a span is a vector space. So, in the case that a line passes through the origin, it’s a vector space. Moreover, points on a line are subsets of $\R^d$, meaning that it’s a subspace of $\R^d$.

>**Note:** A line is a *one-dimensional subspace* as it is spanned by only one vector.

>**Note:** There are infinitely many lines through the origin in $\R^d$. All these are one-dimensional subspaces of $\R^d$.

The animation below shows several lines passing through the origin in $\R^2$.



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

Subspace

Subspace

Examples

Line through the origin