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

# Formulas for determinant
There are three formulas for calculating determinants.

* Determinant by pivots
* Determinant by permutations
* Determinant by cofactors and minors

# Determinant by pivots
The pivot formula we’ll implement in this lesson is based on the property that the determinant of a triangular matrix is the product of its diagonal entries.
$$det(A)= \prod a_{11},a_{22},\cdots,a_{nn}$$


Elimination converts a matrix to an echelon matrix (triangular matrix). Every non-zero entry on the diagonal of the echelon matrix is the pivot. A diagonal entry equaling zero implies the following:

* The matrix has dependent columns. That is, the matrix is singular.
* The product of the diagonal entries is zero. That is, the determinant is zero.

The operations involved in elimination preserve the solution set system. However, not all operations preserve the determinant. As explained in the previous lesson, row sum has no effect on the determinant of the matrix. However, row swap and row scaling operations change the determinant. With each row swap operation, the sign of the determinant is toggled, whereas with every row scaling operation, the determinant is also scaled proportionally. If we track these changes during elimination, we can write the pivot formula as follows:

$$det(A) = P \times \prod a_{11},a_{22},\cdots,a_{nn}$$

where $P$ is the number containing all sign reversals and scaling factors. For computing $P$, we could keep an integer variable that's initialized with $1$, multiplied by $-1$ for each row swap, and multiplied by a factor, $f$, for each row scale. 

The `np.linalg.det()` Python function doesn’t rely on elimination to convert a matrix to a triangular form. For square matrices, we could convert them to their triangular matrix by using only the row sum operation. Let’s look at an example of this:

Implement an algorithm to compute determinants in Python.

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

Determinant by Pivots: A Python Algorithm

Formulas for determinant

Determinant by pivots