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

# What’s a linear combination?
If $\bold{x_1}$ and $\bold{x_2}$ are any two objects and $\alpha_1$ and $\alpha_2$ are any two scalars, then $\alpha_1\bold{x_1}+\alpha_2\bold{x_2}$ is a linear combination of $\bold{x_1}$ and $\bold{x_2}$. We can have several linear combinations of $\bold{x_1}$ and $\bold{x_2}$ by varying $\alpha_1$ and $\alpha_2$.

## Example
Find any three linear combinations of $\bold{x_1} = 
\begin{bmatrix}
3\\
5\\
9 
\end{bmatrix}$ and $\bold{x_2} = 
\begin{bmatrix}
1 \\
2  \\
1 
\end{bmatrix}$.

>**Note:** Objects like column-vectors, such as $\begin{bmatrix}
3\\
5\\
9 
\end{bmatrix}$ , are commonly used to represent points in space. We’ll discuss the vectors and vector spaces in detail in upcoming lessons, but for now, it’s okay to simply view these objects as arrays.

**Solution:** 

$\bold{y_1}=2\begin{bmatrix}
3 \\
5  \\
9 
\end{bmatrix}+3
\begin{bmatrix}
1\\
2\\
1 
\end{bmatrix}=\begin{bmatrix}
9\\
16\\
21 
\end{bmatrix}
\hspace{5mm} \text{where} \quad \alpha _1 = 2, \alpha _2 = 3$

$\bold{y_2}=0\begin{bmatrix}
3\\
5\\
9 
\end{bmatrix}+0
\begin{bmatrix}
1\\
2\\
1 
\end{bmatrix}=\begin{bmatrix}
0\\
0\\
0 
\end{bmatrix}
\hspace{5mm} \text{where} \quad \alpha _1 = 0, \alpha _2 = 0$ 

$\bold{y_3}=-1\begin{bmatrix}
3\\
5\\
9 
\end{bmatrix}+2
\begin{bmatrix}
1\\
2\\
1 
\end{bmatrix}=\begin{bmatrix}
-1\\
-1\\
-7 
\end{bmatrix}
\hspace{5mm} \text{where} \quad \alpha _1 = -1, \alpha _2 = 2$
 

We can generate infinitely many linear combinations from just $\bold{x_1}$ and $\bold{x_2}$. When we choose scalars from an infinite set, such as a set of real numbers, infinitely many linear combinations can be produced from  $\bold{x_1}$ and $\bold{x_2}$.

Below is a code that generates five random linear combinations of two given vectors, $(3, 5, 9)$ and $(1, 2, 1)$.

 Learn the concept of linear combinations, which are a prerequisite of many other vital concepts of linear algebra.

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

Linear Combinations

What’s a linear combination?

Example