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’ve already discussed the inverse of a square matrix in detail and provided multiple examples. In this lesson, we’ll extend the discussion to rectangular matrices, which are matrices that have a different number of columns and rows.


# Rank and invertibility

The rank of a matrix generally answers the question of invertibility. An $m \times n$ matrix $A$ is invertible if it has either a full column rank or a full row rank.

$$r(A) = min(m,n)$$
>Also, for any matrix, $A$,
>$$r(A)=r(A^TA)=r(AA^T) $$



import numpy as np
from numpy.linalg import matrix_rank as r

order = np.random.randint(low=2, high=10, size=2)
m, n = order[0], order[1]
A = np.random.rand(m, n)
print(f'order(A)={A.shape}\nr(A)={r(A)}\nr(ATA)={r(A.T.dot(A))}\nr(AAT)={r(A.dot(A.T))}')

For square matrices, the number of rows, $m$, is equal to the number of columns, $n$. This allows square matrices to have a 
#concept=twosided_Inverse#two-sided inverse#concept#. For rectangular matrices, the condition doesn’t hold. They either have a **left inverse** or a **right inverse**.


# Left inverse

If a matrix, $A$, has full _column_ rank, $r(A)=n$, then it would have a left inverse, $L$, such that:


>A matrix with a full column rank could have multiple left inverses.

One possibility is: 
$$L = (A^T A)^{-1} A^T$$

For a full column rank matrix, $A_{m \times n}$, say, $r(A)=n$, the matrix, $(A^T A)$, will always be square and invertible. We can prove that $L$ will be a left inverse of $A$ as follows:
 
$$\begin{align*}
&r(A) &= n \\
&order(A^TA) &= n\times n \\
&r(A^TA) &= n \\
&(A^TA)^{-1}(A^TA) &= I \\
&((A^TA)^{-1}A^T)A &= I \\
&LA &= I 
\end{align*}$$



Explore the inverse of rectangular matrices and the idea of left and right inverses.

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 Rectangular Matrix

Rank and invertibility

Left inverse