Hi, my name is Anthony Rhodes

and this is Complexity Explorer's tutorial

on Vector and Matrix Algebra.

so this... unit is broken up into three subsections total

in the first subsection we'll be looking at...

domains of numbers, set theory and vectors.

in the second section...

we'll be exploring the notion of matrix

which is really the heart of this unit.

And some of the predominant

algebraic and arithmetic properties of matrices.

At last, in the third section

we will be looking at some applications of matrices
At last, in the third section

we will be looking at some applications of matrices

with an eye towards...
we will be looking at some applications of matrices

with an eye towards...

replications in complex networks and complexity

read large, in particular.

One of the applications we look at, for instance is...

a way to efficiently compute

the power of a matrix,

that's a technique known as diagonalization.

we'll also, in addition to that,

look at some geometric

properties and applications of matrices

known as linear transformations and rotations

And lastly, we will be looking at

applications that relate matrices and matrix computations

to probabilistic models

And that's a technique, or topic, known as Markov chains

So before we begin,

i'd like to first help motivate the study of matrices

by just mentioning briefly a few examples

of places where we can use matrices

to study interesting phenomena

and applied math in science

so, for instance, one of the all-time

classic applied problems really in the history of

of math and science

has to do with curve fitting

what i mean by that is if I perform an experiment
has to do with curve fitting

what i mean by that is if I perform an experiment

and collect data

I would like to

for the purposes of prediction in inference

to build a model that best fits that data

In a general setting, this is known as

the problem of regression.

So as it turns out

in learning this best-fit curve

i can include the relationship between the data

and the curve I'd like to build,

or the model i'd like to build

using matrices

And those equations are very well known and well studied

And they're known as 'the normal equation'.

You see oftentimes matrices

as representing operations

or quantum computations

and... statistical physics, for instance

a matrix can include relationships

in a vector field

if I want to know how particles in room

interact, i can do that.

actually, as it turns out, via matrix multiplications...

And, of course, network theory and graph theory

replete with examples that apply

ideas related to matrices.

and let's cite just a few of those

so, for instance,

if I'm in city A

and I'd like to find the most efficient route to city B

and we can think of this as, maybe, the code to Google Map problem

which we'll explore later
and we can think of this as, maybe, the code to Google Map problem

which we'll explore later

I can query Google Maps or some similar algorithm

get a pretty decent result as to the most efficient

way or efficient route from

city A to city B.

In general, that algorithm, or variant of it is

known as Dijkstra's algorithm

As another example we can think of

in fact, the Internet itself

as representing a complex network or graph

where the nodes, or vertices, in the graph

are computers or servers that connect to

via some link.

So we might wonder when studying a complex network

like the Internet, are some of the nodes

more important, or some of the servers more important, than others.

And we might wonder such a thing because, well

well, if we lost some of those servers,

is it possible that we have catastrophic

failure of the Internet, or some such network

Because we don't want that, so we'd like to take the necessary precautions.