# 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.
So 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
<i>x</i> + <i>y</i> = 5
and <i>x</i> - <i>y</i> = 1.
And that, 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 the system
have a solution or not?
Now, in this simple case,
of course, we have
that <i>x</i> is 3 and <i>y</i> is 2,
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.
In any case, for some people,
their definition
of linear algebra is
linear algebra is the study
of linear systems of equations.
Other people have
a more geometrical view
of the subject,
and they'll say that
Linear algebra is the study
of geometric transformations.
For example, here in this figure
you have a reflection about
the line <i>x</i> = <i>y</i> in the plane.
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 <i>x</i> = <i>y</i>
is kind of invariant or unchanged
for this transformation.
Some people will tell you
that linear algebra
is a study of
linear geometrical transformations.
But of course
there is a third viewpoint,
which is quite popular
these days,
and that's the notion
of matrices and determinants.
A 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 the 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,
and so doing linear algebra
is actually a very fruitful way
of solving problems
in the modern world.
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 used
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 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.
The question is:
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 a 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 physicists 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.
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 it 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 their 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 exercises and quizzes
going on to accompany the lessons.
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.
My main goal in all this is
to show you that
linear algebra is really easy,
and I hope that when you're done,
you will feel that
You've learned linear algebra
in a useful way
and you'll be able to
apply it in your own work,
whatever subject it is
that interests you.
So the question is:
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.
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.
Really 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 teacher,
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 the study of linear equations
and geometric transformations.
The modern approach to the subject
will use matrices, determinants,
and at a more advanced level,
it will also use vector spaces.
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,
we'll begin the course formally
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.