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

# Multi-target data sets
It’s common in real applications to predict more than one target from given features. The 
#concept=UCI_Dataset#data set#concept#
 we’ve discussed in the previous lesson comes from a similar scenario. We’ve already approximated a single target given its multiple features. In this lesson, we’ll predict both Y1 (heating load) and Y2 (cooling load), given X1 to X8 features. We can do so using **multi-target linear regression**. Formally, if $\bold{x_1},\bold{x_2},...,\bold{x_d}$ are the feature columns and $\bold{y_1},\bold{y_2},...,\bold{y_c}$ are the target columns, we can define data matrix as $A_{n \times d} = \begin{bmatrix}\bold{x_1}&\bold{x_2}&...&\bold{x_d}\end{bmatrix}$ and the target matrix as $Y_{n \times c} = \begin{bmatrix}\bold{y_1}&\bold{y_2}&...&\bold{y_c}\end{bmatrix}$, where $n$ is the number of data points. 

The linear system can then be modeled as: 

$$AW=Y$$ 
where, $W=\begin{bmatrix}\bold{w_1}&\bold{w_2}&...&\bold{w_c}\end{bmatrix} \text{ is a }d \times c \text{ matrix of parameters.}$

# Objective
The goal is to find $W$ that minimizes the objective $\|AW-Y\|_F^2$,  where the subscript $F$ represents the **Frobenius norm**.

## Frobenius norm
A ***Frobenius norm*** of any matrix is defined as the square root of the sum of squares of all its elements. Formally, for any matrix, $A_{m \times n}$, the Frobenius norm is $\|A\|_F=\sqrt{\sum_{i=1}^m\sum_{j=1}^na_{ij}^2}$.
### Frobenius norm in Python 
We can use ```np.linalg.norm``` to compute different types of norms of vectors and matrices. For matrix input, the default type is the Frobenius norm, and we can also set it explicitly as a parameter, ```ord='fro'``` as described in the code below:

Learn to implement linear regression on a real data set with multiple targets.

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

Multi-target Linear Regression

Multi-target data sets