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

# Functions in data science
Functions are generally not restricted to having only numbers, arrays, matrices, or #key# tensors: Generalisation of scalars and vectors #key# as inputs. The same is true for a function’s outputs and parameters. However, data science focuses on functions that consist of inputs, outputs, and parameters that are numbers, arrays, matrices, or tensors only. This is because the objects of interest, including images, text, audio, videos, and so on, are all #concept=NumericalFeatures# numerical features #concept#.  Let’s look at a few examples to understand this better.

## Face recognition




A face recognition system takes the image of a face as input and outputs a number describing the identity of the face in the image. In this case, the face recognition system is a _function_, and the face image, usually represented as a matrix, is _input_. The function’s _output_ is a _scalar_ pointing to the position of the person’s name in a list. Thus, this function maps a matrix to a number.

## Speech recognition

The figure above illustrates a typical speech recognition system. The input of the system is in the form of raw audio, usually represented as an array in the system. The output depends upon the speech processing task at hand. For instance, the textual output is usually expressed as an array in speech-to-text conversion. 

## Object detection




In an object detection system (depicted above), the input is an image, whereas the output is an array of bounding boxes and object identities in that image. Both the image and bounding boxes can be represented as matrices.
# Function approximation in data science
As described in the examples above, the output of a real-world system is an unknown function (say, $f$) of its input. Our goal in data science is to estimate this function reliably. We don’t know anything about the function except for a few sample inputs and corresponding outputs called ***training data***. Let's also assume the type of underlying function (say, a a second degree polynomial)  based on some data analysis. We still have to estimate its parameters to best relate the inputs to the corresponding outputs (because there may be several second degree polynomials, depending on the parameters).

More formally,  considering the given inputs, $x_1,x_2,...,x_n$, and corresponding outputs, $y_1,y_2,...,y_n$, our goal is to find the best approximation of a function, $f$, having parameters, $w$, such that:
$$f_w(x_i)=y_i, 1 \le \forall i \le n$$

An exact function may be hard to determine or may not even exist. So, based on some criterion (loss), the function is approximated using estimated parameters. We’ll discuss this in more detail later.

# How does linear algebra help?
_Linear functions_ can well approximate the relationships between most underlying objects. Even if the actual relationship isn’t linear, it can be approximated through several linear functions. This is precisely what happens in deep neural networks too. Linearity makes many things tractable and straightforward. Linearity is useful in function approximation, particularly in data science.


Learn the use of functions in data science.

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

Functions in Data Science

Functions in data science