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Hello and welcome to Linear Algebra!
This is actually not the first linear algebra
video that I'm recording.
I have already recorded and posted over
160 individual videos.
In fact, you can see the remnants of
the last one I just recorded on
the blackboard behind me.
And all along I've been thinking about
the best way to do this introduction.
There's a number of things I could
talk about in this introduction.
I could give you a few of
the thousands upon thousands of
applications of linear algebra.
Or I could talk about the importance
of linear algebra in our digital age.
Before we had computers, our primary tool for
describing the world around us had been calculus.
But now that we have computers,
linear algebra may be even more powerful than calculus.
I could also talk about the wonderful
algorithmic and logical aspects of linear algebra.
Or I could talk about
how I see linear algebra as
the place where algebra and geometry meet,
lend each other forces,
and become much more powerful
than either subject had ever been on its own.
I could talk about any of those things,
but I will talk about none of those things.
Instead, I will talk about one mystery
whose unraveling requires
all fundamental aspects of linear algebra.
And this mystery is about the
difference between how our eyes perceive light
and how our ears perceive sound.
You will agree with me that
every moment of the day, we have
a chance to see something breathtaking.
Just look at some of these images!
This is possible thanks to
one extraordinary organ that we have
and that of course, is the eye.
The eye is very complex to say the least
so we will focus on one particular aspect
of one particular part of the eye:
the retina
-- seen in this image as a red line
around the perimeter on the inside of the eye.
The retina is home to the light sensors
called rods and cones
and it's thanks to their great numbers
that we're able to see such striking images.
In total, there are around 100 million of them.
You can think of this number as
the resolution of the eye.
By comparison, the most advanced monitor today
has 15 million pixels
or less than 1/6th the resolution of the eye.
But what about the ear?
How many sensors does the ear have
that allows it to perceive such beautiful music?
Actually, let me ask you the same question
in a more complex situation.
Suppose you're listening to this composition
by Antonio Vivaldi
played by the Philadelphia Orchestra
in Yankee Stadium
with 50,000 fans cheering the orchestra on.
How many sensors does the ear need
to take in such complexity?
Here
Pick the number from the list
that you think is about right.
The answer might surprise you
because it's 1.
There is a single signal that enters your ear
and therefore, takes a single sensor to receive it.
Here is what happens:
All the sounds simply get added together
and it is their sum that enters your ear.
But how is that possible?
How can one signal
contain such richness
and how is your ear
able to hear the multiple
instruments and voices individually
from just one signal?
Well...
This is the mystery that linear algebra will eventually help us solve.
Meanwhile, I will show you a demo
that will give you a glimpse into
how a single signal
can contain numerous pieces of information.
And what kind of analysis
needs to be performed in order to
extract those individual pieces.
All right!
So here is what we are going to do now.
I will play three notes
simultaneously on this piano
and I won't tell you what those notes are
But then we'll look at the resulting audio signal in the computer
and by analyzing that signal
we'll try to determine
what those notes were.
So that's the task.
Now because this is a digital piano,
when I play those three notes at the same time
its internal logic
will add the three signals together
and then its speakers will
play the resulting sum.
Had it been a real piano,
then the three notes would sound individually.
But then the laws of physics
would add up the three signals together.
And in either case,
this microphone receives the resulting sum.
The task we're faced with
is decomposing that one signal
into the individual notes
so the task we're faced with
is that of decomposition
which is taking the resulting sum
and determining what elements went into that sum
and in what proportion.
The problem of decomposition
is the first and perhaps
the biggest topic in linear algebra.
So it's a great place to start.
So let me play those three notes at the same time,
record the audio,
and then we'll look at that signal
on the computer.
So here they are:
[Piano chord]
Alright, so now let's walk over to the computer
and look at the result.
So now we're looking at the signal we just recorded
and please notice it's not three different signals
representing three different notes.
It is a single signal
representing all the notes at once.
And it's our task
to somehow see the individual notes
in this one signal.
And we're currently fully zoomed out
and at this level of detail
we can't say much
except perhaps that the sound starts out loud
and then gradually dies out.
And we're also seeing
very subtle oscillations in volume
known as beats.
If you listen to the sound very carefully,
you will be able to hear those beats.
But at this zoom level,
we do not have nearly enough detail
to even begin determining
the notes that were played.
So let's zoom in on a very small part of our signal
right around here
that will give us sufficient detail
to try and solve the problem we're facing.
Having zoomed in on 4/100th of a second,
we're finally seeing the complexity involved.
And of course it's this complexity
that's responsible for storing such rich information
in a single signal
So let's begin to analyze
what we're seeing.
What we're seeing is a signal
that's somewhat periodic
and that's characteristic of musical tones.
There are really two major features
that our eye easily picks up.
First of all, we're seeing these
larger oscillations
of which there are
1
2
3
4
5
6
7
and on top of these larger oscillations,
we're seeing smaller, more frequent oscillations
that we'll count in a moment.
But right there
we just did our first bit of decomposition.
We have looked at one thing
and within that one thing
we saw a sum of two different things.
And those two things
were the larger, less frequent oscillations
and the somewhat smaller,
more frequent oscillations.
So that, in a nutshell, is Decomposition --
seeing a single thing
as a sum of two different things.
Let us now try to figure out what notes we're looking at.
And that of course is determined by the frequencies of these oscillations.
Now, there are seven of the larger oscillations
in the 4/100th of a second,
which corresponds to 175 oscillations/second.
In order to figure out what note this frequency corresponds to,
let's refer to this...
...Wikipedia article on the frequencies of musical tones.
We find out that 175...
corresponds...
...to the F in the 3rd octave.
Great!
We have just determined
one of the three notes.
In order to identify another note,
we have to determine the frequency
of the smaller oscillations.
Would you agree with me that
there are roughly 6 of the smaller oscillations
for each of the larger ones?
That means the frequency of the smaller oscillations...
is...
175 (the frequency of the larger oscillations)
times 6
or approximately 1050
which corresponds...
...to the C in the 6th octave.
We have therefore been able to determine
two of the three notes.
And I think that's about
all we can do with a naked eye.
So how well did we do?
The answer is...
not bad at all!
Let me reveal the notes that I had originally played.
They were indeed
the F in the 3rd octave,
[Plays F]
followed by an A flat,
[Plays A flat]
and a C
[Plays C]
Here they are all together:
[Plays chord]
So we nailed this F
[Plays F]
Instead of this C,
[Plays C]
I picked out this C
[Plays high C]
which was a small mistake
-- they're very similar notes.
[Plays both Cs]
And we did miss the A flat:
[Plays A flat]
But 2 out of 3 notes really isn't bad
given how simplistic our approach was.
And it certainly demonstrated
the linear algebra idea of Decomposition
which will occupy
nearly a 1/3rd of our entire linear algebra course.
Just imagine what we'll be able to accomplish
in terms of recognizing notes,
once we learn a little linear algebra
and unleash its powerful algorithms
upon this problem.
Meanwhile, we really are
off to a great start.
So there you go!
I hope that you enjoyed that presentation
and that you're excited about this subject
of linear algebra.
Not that that presentation could make you go from
not excited to now excited
but I hope that you like our informal approach
and our emphasis on applications.
And if that presentation didn't make total sense to you,
there is nothing at all to worry about
because we are going to start
at the very beginning.
So welcome aboard,
and let's learn some linear algebra!