# MOOC_LinearAlgebra_Lesson01

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Hello and welcome to
Introduction to Linear
Algebra with Wolfram U.
My name is Devendra Kapadia
and I'm the Manager of Calculus
and Algebra at Wolfram.
But I also enjoy teaching
mathematics, and, in fact,
I have taught mathematics
at high school and college
level for many years.
It is a great pleasure to
share my ideas about linear
algebra with you in this course.
This is the first lesson
of the course, let’s begin
by asking the question:
what is linear algebra?
Well, linear algebra
means different things
to different people.
For some people linear
algebra means linear systems.
Here is a linear
system of equations.
You have the equations
x+y=5 and x-y=1.
And there, of course, is
just a pair of straight
lines in the plane, which you
can see graphed over here.
Given the system, we have
some key questions to ask.
For example, does this system
have a solution or not?
Now, in this simple case,
of course, we have that
x is three, and y is two,
and that's the solution.
But in some other cases, you
may not really have a solution
at all, or it might be quite
hard to find a solution.
So, we have a notion of
an approximate solution.
So, either you have an
exact solution or you have
an approximate solution.
At any case, for some
people, their definition of
linear algebra is, linear
algebra is the study of
linear systems of equations.
But other people have a
more geometrical view of a
subject and they’ll see that
linear algebra is the study
of geometric transformations.
For example, here in this figure
you have a reflection about
the line y=x in the plane.
And this, of course, is a
linear transformation because
the image of a line is,
again, a line and given such a
transformation, you could ask:
find an algebraic description
of this transformation, or
you could ask: are there any
lines that are left unchanged
by the transformation?
For example, over here you
can see that the line y=x is
kind of invariant or unchanged
for this transformation.
So, some people who tell
you that linear algebra
is a study of linear
geometrical transformations.
There is a third viewpoint,
which is quite popular these
days, and that's the notion
of matrices and determinants.
So matrix is just a
rectangular array of
numbers in rows and columns.
So here's a matrix,
it's got two rows.
First row is 5, 4,
second row is 6, 7.
And 2 columns, so
that's a 2 by 2 matrix.
A system of linear equations
and transformations can actually
be represented using matrices.
Now, a matrix is a collection
of numbers, that's a huge
collection typically but,
with every matrix you
have its determinant.
The determinant is
just a number that's
associated with the matrix.
For example, in this case, you
have the determinant as 11.
That comes out by doing
5*7=35, 6*4=24, subtract
them and get 11, and this
determinant, although it’s
just a number, it encodes, it
contains important information
about the system of equations
or the transformation
that you’re studying.
So in the modern world, a
nice way to think of linear
algebra, is that it’s a study of
matrices and their determinants.
And of course, that's useful,
because in this day of computers
we can store matrices quite
easily with powerful computers.
So, doing linear algebra
is actually a very fruitful
way of solving problems
in the modern world.
Now of course, there's a
long history behind the
subject but let me just give
you a brief history of the
subject of linear algebra,
as it relates to this course.
So, of course, the Chinese
and other civilizations
knew about linear algebra.
But to me, the course
has history going back
to roughly 1750 when
Gabriel Cramer published
a study of determinants.
And in fact, he published
a rule for solving
systems of equations.
And a little later, 1800, the
famous German mathematician
Carl Gauss actually talked
about elimination to solve
equations and the least squares
method, which we'll talk
about later in the course.
Now, Gauss was a genius
but his real interest in
this case, was to try and
do geodesy and apply these
ideas to real world problems.
But the mathematical theory
of linear algebra really
began around 1850, when James
Sylvester use the term matrix or
“womb” for an array of numbers.
Sylvester and his friend, Arthur
Cayley, worked on matrices,
and Arthur Cayley actually
published his work on the
theory of matrices around 1855.
Sylvester and Cayley
were great friends.
In fact in those days,
it was hard to make a
living doing mathematics.
So Sylvester was an actuary,
he worked in insurance.
Cayley was a lawyer, and
they talked about law in
the day, and talked about
mathematics in the evening.
Together, they set up
a lot of what we call
Linear Algebra Theory.
A little later, in 1888,
Giuseppe Peano actually
talked about vector spaces.
In the modern mathematical
world, vector spaces are
very important, but in this
course we will not worry
too much about them, we’ll
mostly work with matrices
and determinants as taught to
us by Cayley and Sylvester.
what happens today?
Well today, people all
over the world use linear
algebra in different ways.
In different professions
there are their own
uses for linear algebra.
For example, if you do
mathematics, you might want
to study the geometry of
surface, and then you’d use
linear algebra to actually
make a system of coordinates
in which you can do your
matrix manipulations.
Engineers use linear
equations to study the
motion of fluids, to design
airplanes and such things.
Data scientists will have
their own linear models to
make predictions about data.
In fact, our data science
is a big driver for
linear algebra these days.
Now, if you study physics,
then physicist will use linear
algebra to study quantum
mechanics, the study of atoms.
In fact, these days, they
also use linear algebra to
build quantum computers.
And finally, it's not
just the sciences.
Economists have always used
linear algebra for doing things
like, input-output matrices,
and in fact, a lot of modern
linear algebra came around
because of these applications.
So not only have the
applications used linear
algebra, but they actually
have driven the modern
development of the subject.
So linear algebra is very
much an applicable subject
in the modern world.
Now, of course, you might
ask the question: why should
I study linear algebra?
Well, I'll give you
a few reasons for
studying linear algebra.
The first of all, it has
very broad applications
in the real-world.
We just saw that.
If you apply for certain
kinds of stem degrees, it's an
essential course which you'll
need to cover at some point.
In fact, in some cases,
they may even ask you, how
good is your linear algebra
for advanced degrees.
Now, even if all that were
not true, I would say that,
learning linear algebra is a
great thing, because linear
algebra is a major intellectual
endeavor, and to study is a
major intellectual achievement.
So, I’d say that to study
linear algebra is to master
one of the most important
branches of modern mathematics.
And I strongly urge you
to do it with this course.
So now let me tell you a little
bit about the course itself.
The main goal of this course
is to give you a thorough
introduction to linear algebra.
And to do that, I'll begin by
talking about linear equations
and linear transformations.
And then we'll talk about
matrices and the determinants
and then we'll move on to
more advanced topics like
eigenvalues and other things.
Now, to make sure you understand
what's going on, we will have
excellent quizzes going on to
accompany the lessons, and as
far as proofs are concerned,
there will be no formal
proofs in the course and we'll
focus on understanding the
subject without any proofs.
Now, my main goal in all this is
to show you the linear algebra's
really easy, and I hope that
when you're done, you will
feel that you've learned linear
algebra in a very useful way.
And you’ll be able to
apply it in your own work.
Whatever subject it
is that interests you.
who is this course for?
Well, you might be an
ambitious middle schooler or
high schooler who wants to
learn advanced mathematics,
become an engineer, become
a data scientist, whatever.
I would say to you that,
to study linear algebra
is a great way to start
learning more mathematics.
So this course is really
designed for people
who are learning linear
algebra for the first time.
Or you might be a busy
professional, who needs to get
some knowledge of linear algebra
to, let’s say, take a course on
machine learning or whatever,
and then this course should
be a nice refresher for you.
I think that this course is good
for anyone who wants to know how
the Wolfram Language can be used
to learn and do linear algebra.
So, if you love mathematics and
you love programming, then this
course is certainly for you.
Whether you are a
student, a professional,
a teach, or anyone else.
I think this course is meant
for a wide variety of people
and I’ve tried to keep
it as simple as possible.
That brings me to the
end of this introduction.
To summarize, linear algebra
is a study of linear equations
and geometric transformations.
The modern approach to
the subject will use
matrices, determinants.
and of course, at a more
advanced level, it will
also use the vector spaces.
Now, to know linear algebra
is important because it's
an essential tool for
students in many fields.
That's the broad
introduction to the course.
In the next lesson will begin
the course formerly with an
introduction to linear systems.
Thank you very much.
I hope you enjoyed
this introduction.
I do hope you'll pursue this
course with great sincerity
and wish you good luck.
Thank you.