In this lecture I want to give you a few examples of linear algebra used in practice.
This will give you a general idea of what people do with linear algebra and why Linear Algebra is becoming
more and more important and more more ubiquitous in all areas of science and engineering.
Maybe you'll even recognize some areas where you didn't think linear algebra was so useful or so important.
The first example will be data analysis and big data.
Now first of all data are typically stored as matrices like what you see here.
Storing data as a matrix means that I can represent this entire dataset using a single letter.
In this case for example the letter Y now most of the analysis performed in statistics and in machine
learning are represented using vectors and matrices and the analysis are implemented using matrix operations.
For example one way to try to understand these structures and the feature of a large dataset is to build
a model of the data.
A model is a set of equations that captures what you assume are the underlying dynamics of the important
features in the data.
The model is then fit to data by estimating values of free parameters.
So here you see an example of a model expressed as a set of equations.
Now in linear algebra these equations are then re expressed in a more compact form using matrices and
vectors.
Here is the matrix equation that represents this system of equations.
In other words the model of the data you can see that this is much more compact than this representation
then Fitting the model to the data involves computing this solution that you see here.
Now this might look confusing and it might look weird and intimidating but don't worry.
By the end of this course you will understand exactly what this equation means where it comes from both
algebraic Lee and geometrically you will know how to derive it from scratch using several different
methods and you will even be able to solve equations like this by hand.
And of course more importantly you will know how to implement equations like this in MATLAB and in Python.
A second example comes from computer graphics.
The idea of computer graphics is to have an image that you can move around on the screen translations
and rotations in three dimensions can be implemented by multiplying the coordinates of the image by
this three by three matrix.
This is called a rotation matrix and it's a really compact way and an efficient way to manipulate the
location and the perspective of pictures on the screen.
Actually many modern computer graphics applications and algorithms rely on something called quite turn
ins which are vectors containing four dimensional complex numbers.
Now these quarter earnings are a little bit more computationally efficient but the principle is exactly
the same as this kind of translation and rotation matrix A third example is from graph theory graph
theory is a mathematical framework that is used to understand networks like this graph theory is crucial
to a wide range of topics including air traffic control the internet social media networks and even
the brain.
Many problems in graph theory can be solved by using matrices to represent the information in the graphs
of the nodes and all the con activities across the different nodes.
Here you see an example.
Some of the core features of the connectivity in this network can be represented using a matrix that
would look something like this.
Now I could go on for hours and hours giving you a million different examples of how matrices and principles
of linear algebra are used to solve problems in science and engineering in finance and computational
biology and so on.
Linear Algebra is also a prerequisite for multivariate calculus and differential equations.
But you don't even have to care about these practical applications to find linear algebra.
A fascinating topic to learn there is something intrinsically satisfying about linear algebra matrices
are so compact and yet they contain so much rich information and the link to geometry is fascinating
and beautiful.
I tried to highlight the links between linear algebra and geometry as often as possible.
In this course Linear Algebra is also the key backbone the computational workforce that underlies many
modern algorithms in big data science machine learning artificial intelligence and statistics.
Therefore linear algebra will help you improve your programming skills while also learning about modern
algorithms and modern data science.
In this course you will learn how these kinds of pictures are visual representations of matrices and
you'll learn how to extract features or information out of the geometry to help you understand modern
computational algorithms.
Regardless of whether you taking this course because you want some supplementary learning material for
a class at school or university because you're trying to improve your chances of getting a job or getting
into a good program because you want to improve your programming skills using linear algebra or because
you were just curious about one of the most important modern topics in mathematics I hope you find the
rest of this course to be useful informative and enjoyable.