Welcome to the matrix.
So in this lecture,
we'll just give a little feel for linear algebra
because this is not really a linear algebra course.
Now we said that there's an object in
math in linear algebra actually called a matrix.
For instance, what we see here, 1001 is the identity matrix.
And we said the identity matrix
doesn't do anything. And indeed, we can think of a matrix as a function.
So it's taking Homer Simpson's face
and just
sending it to Homer Simpson's face, it's doing nothing.
But if I start changing values, for instance, let's put
a two over here,
then it scaled Homer
by two in the X direction.
If we put a two
over here,
we also get a scale by two in the Y direction.
Now I'm gonna undo all that.
So we have the identity again. And now let's put a one in the upper right hand corner.
This is called a shear transform. It's kind of like italics
and I can increase this value
and I can also go in the negative direction
and it's always good to try and explore.
OK, let's try something else.
So let's put, say
negative 1.4 over here.
And I'm gonna put, say
1.4
and 1.4.
And this is a rotation
of uh 45 degrees.
So there is such a thing, OK. This program is not really uh cooperating with me.
I'm trying to enter 1.4 over here.
So it's approximately 45 degrees.
But in general, there's such a thing called a rotation matrix
where you have
cosine
sine, negative sine
cosine
wherever you put,
the negative kind of decides if you're going clockwise or counterclockwise.
Now it turns out that these matrices
many times they model,
they model something in nature, they model some phenomenon.
And every time you multiply by a matrix
you have some process that's iterating over time.
So the more you multiply by a matrix, for instance, you could have
a description of a population,
right. So a population
of red
blood cells using linear algebra. This is one of my
favorite examples,
one of many.
So using linear algebra in biology
and this is used in programming and economics and math.
So
they're talking about a population of red blood cells in general,
red red blood cells, you know new ones are born some die
and they created some model involving matrices.
Now, the thing that's important to understand here
is this letter M,
this letter M describes a model OK. A model in nature.
It might be an approximation but it's some kind of model.
Now
we are interested in the nth power of that model
because the nth power expresses
n iterations like say
a million seconds into the future. So N equals to a million.
Now it turns out that powers are very difficult to calculate
of a matrix unless you're dealing with a diagonal matrix.
So one of the central things that we do in math is try
to replace in linear algebra is to try to replace a matrix M
which is roughly speaking
a diagonal matrix D,
they use the word similar and this is a precise concept.
So we're very, really interested in
the process of of diagonalization a matrix.
And also matrices are used everywhere
in machine learning and artificial intelligence
in a sense, matrices des
describe very simple functions in a sense
functions that we understand very well and that we can calculate very quickly.
Now
in this paper, they say this model is an approximation.
So they might have to make some changes for it to be a little bit more realistic.
OK. So that's just trying to sell you linear algebra
in about a minute.
So in the next lecture, we'll start seeing
matrix multiplication, scalar multiplication
addition
of finding the trace and determinant of a
matrix which are all very important numbers.
They're kind of invariances
that describe your matrix just like the average is
an important num number involving a list of numbers.
So a determinant
um trace
and also diagonalization a matrix. These are very important concepts
and I'd be happy to expand upon them even though this course isn't really proper a
linear algebra course.
OK. So that's just to get if you feel
of linear algebra
and at some point, linear algebra and calculus really combine
to solve real world problems, especially in artificial intelligence.
Great.
So have an amazing day
and see you next time.