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

## Why polynomial regression?
Until now, we’ve assumed a linear relationship between features and targets. What if the assumption of linearity doesn’t hold even approximately? The figure below shows that the relationship between a single feature, $x$ and a single target, $y$ isn't linear in the data set.

$$ f(x) = \hat{y} = \sum_{i=0}^d w_i x^i$$

## Assuming degree to be known
Let’s look at how we can estimate the parameters, $\bold{w}$, if we assume, for the sake of simplicity, that the degree of the polynomial to be fit is known.

### Example of polynomial with degree 3
Let's take a simple example with only one feature, $x$, and a single target, $y$. Let's also assume that $y_i = w_0+w_1x_i+w_2x_i^2+w_3x_i^3$. We can notice here that
$$\begin{bmatrix}1&x_1&x_1^2&x_1^3\\1&x_2&x_2^2&x_2^3 \\ \vdots & \vdots & \vdots & \vdots\\1&x_n&x_n^2&x_n^3\end{bmatrix}\begin{bmatrix}w_0\\w_1\\w_2\\w_3\end{bmatrix}=\begin{bmatrix}y_1\\y_2\\ \vdots \\ y_n\end{bmatrix}$$

​
This is a linear system with a data matrix consisting of transformed features. The transformation of features from $x_i$ to $\begin{bmatrix}1&x_i&x_i^2&x_i^3\end{bmatrix}$ is straight forward. After the transformation of features, it’s just **linear regression**.



The polynomial features can be easily computed using `sklearn` as follows:
```python
from sklearn.preprocessing import PolynomialFeatures
pf = PolynomialFeatures(degree = 2)
x_tf = pf.fit_transform(x)
```
## Multi-target linear vs. polynomial regression
We’ve already seen multi-target regression. In the figure below, let’s see the results with polynomial regression of `degree = 2` on the same dataset.

# Why polynomial regression?
Until now, we’ve assumed a linear relationship between features and targets. What if the assumption of linearity doesn’t hold even approximately? The figure below shows that the relationship between a single feature, $x$ and a single target, $y$ isn't linear in the data set.

$$ f(x) = \hat{y} = \sum_{i=0}^d w_i x^i$$

# Assuming degree to be known
Let’s look at how we can estimate the parameters, $\bold{w}$, if we assume, for the sake of simplicity, that the degree of the polynomial to be fit is known.

## Example of polynomial with degree 3
Let's take a simple example with only one feature, $x$, and a single target, $y$. Let's also assume that $y_i = w_0+w_1x_i+w_2x_i^2+w_3x_i^3$. We can notice here that
$$\begin{bmatrix}1&x_1&x_1^2&x_1^3\\1&x_2&x_2^2&x_2^3 \\ \vdots & \vdots & \vdots & \vdots\\1&x_n&x_n^2&x_n^3\end{bmatrix}\begin{bmatrix}w_0\\w_1\\w_2\\w_3\end{bmatrix}=\begin{bmatrix}y_1\\y_2\\ \vdots \\ y_n\end{bmatrix}$$

​
This is a linear system with a data matrix consisting of transformed features. The transformation of features from $x_i$ to $\begin{bmatrix}1&x_i&x_i^2&x_i^3\end{bmatrix}$ is straight forward. After the transformation of features, it’s just **linear regression**.



The polynomial features can be easily computed using `sklearn` as follows:
```python
from sklearn.preprocessing import PolynomialFeatures
pf = PolynomialFeatures(degree = 2)
x_tf = pf.fit_transform(x)
```
# Multi-target linear vs. polynomial regression
We’ve already seen multi-target regression. In the figure below, let’s see the results with polynomial regression of `degree = 2` on the same dataset.

Learn how linear regression can be used to build a polynomial regressor.

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

Polynomial Regression

Why polynomial regression?

Assuming degree to be known

Example of polynomial with degree 3