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!