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HERBERT GROSS: Hi.
We've sort of arrive
at D-day in our course;
that in a manner of speaking,
everything that we've done up
to now has been a rehearsal.
That today we are
going to come to grips
with what the course
is all about: namely,
a real-valued function of
several real variables.
And before getting into
that, what I'd like to do
is to review very briefly
what we've talked about
so far in terms of
functions, since we've
introduced the vector notation.
Namely, what we've
mentioned is, is
that our function
machine could have
either a vector or a
scalar as its input,
and either a vector or
a scalar as its output.
Part one of our course
centered around the idea
where both the input and
the outputs were scalars.
Whereas, the previous
block of material
concerned the case where
our input was a scalar,
and our output was a vector.
Today what we're
going to discuss
is the situation that occurs
when our input is a vector,
and our output is a scalar.
And I call today's lecture
"n-dimensional Vector Spaces."
Eventually, we'll discuss
vector spaces in more detail.
For the time being, I
simply want to set the mood,
and hopefully by the
time I'm through,
show you a rather peaceful
coexistence between the worlds
of the new mathematics,
and the worlds
of traditional mathematics.
In fact, in many of
our topics that we're
going to tackle in
this block of material,
we will give both
points of view.
But to set the stage
properly-- to get
into the idea of what
a vector space is all
about-- and if that
word frightens you, just
don't worry about
for a minute or two.
Worry about it after that,
but let's just get started.
Let's consider the
situation where
I have a function machine
where my input is a vector
and my output is a scalar.
As an example, let me just
generically let v represent
the vector whose
form is x*i plus y*j,
using i and j components.
Let me define f of v to be
pi times the square of the i
component times the j component.
And let's not worry
right now about why I
picked this particular recipe.
Let's simply observe that
once this recipe is chosen,
f is a function which maps a
two-dimensional vector here,
in i and j components, into
a number of pi x squared y.
Just to illustrate
this recipe, notice
that if our vector
were 3i plus 4j,
the output of the f machine--
if this were the input--
would be what?
Pi times the square of the first
component-- the component of i,
that's 3 squared-- times the
component of j, which is 4.
And that leads to 36*pi.
By the way, observe that
order does make a difference,
namely if I reverse the
roles of the coefficients,
and feed the vector 4i
plus 3j into my f machine,
the output would be
pi times 4 squared--
the square of the i
component-- times 3,
and that would be 48*pi.
Notice also that we have
allowed, already in our course,
the abbreviation that a comma
b would represent a*i plus b*j,
or a comma b comma c would
represent a*i plus b*j plus
c*k.
The idea is that if we
now apply this shorthand
notation to these two vectors,
what we could say is what?
f of this vector, in other
words, f of 3 comma 4 is 36*pi,
whereas f of 4 comma 3 is 48*pi.
And that's exactly
what we mean when
we say that we may treat
a two-dimensional vector
as an ordered pair.
You see, it's, not only
a pair but the order
does make a difference.
Both in what the
vector is and what
the output of the f machine is.
OK?
Hopefully, let's
say, so far so good,
and let's tackle now a rather
completely different problem.
What I'd like to do
now is the following.
Let's consider the cylinder--
the right circular cylinder--
the radius of whose base is
x, and whose height is y.
Notice that the volume of this
cylinder is pi x squared y.
And if I use the same f
that we used previously--
in other words,
the same f that we
were talking about over here--
notice that another way--
see how is f defined?
Given that the input was x
comma y the output was pi x
squared y, notice that
the volume is f of x, y.
In particular, if I want
the volume of the cylinder,
the radius of whose base is
3, and whose height is 4--
see x is 3 and y is 4--
what I really want
is f of 3 comma 4.
That's pi times 3
squared times 4.
f of 3 comma 4 is 36*pi.
Now I'd like to
pause for a second
again, and return to
our earlier remark.
Namely, if I look at this,
and if I look at this.
Notice that these two
expressions are identical.
I cannot tell the difference
between whether I'm looking
at the vector 3i plus 4j,
or whether I'm looking
at the cylinder, the
radius of whose base is 3,
and whose height is 4; whether
I look at this equation,
or whether I look
at this equation.
The difference is that in the
first case somehow or other,
it was quite natural
to think of 3 comma
4 as being either an
ordered pair, or an arrow.
In this case,
however, my contention
is that when we
think of the radius
of the base of the
cylinder, and the height,
we do not tend to think
in terms of arrows,
but rather in terms
of ordered pairs.
In other words, the ordered pair
x comma y, in the expression
f of x, y, need not be
viewed as an arrow, but as
an ordered pair.
And an ordered pair
is called a 2-tuple.
This leads to a
generalization that I
think is rather
important, and I think
you will see in a moment,
where the idea of this approach
comes into functions of
several real variables.
The topic I have in mind is
something called an n-tuple.
And let me read into that
rather gradually as follows.
Without giving you a specific
physical example-- meaning
I'll give you an illustration,
but leave the numerical amounts
out.
Quite possibly if I'm studying
temperature in a room,
the temperature will
in general what?
It will depend on what
position I'm at in the room,
and also at what time I
measure the temperature.
It's fair to assume that
in many applications
the temperature is some
function of the four
independent variables
x, y, z, and t,
where x, y, and z are
the Cartesian coordinates
of three-dimensional space,
and t represents time.
What I'm driving at is I can
now visualize this in terms
of my function machine again.
Namely to compute T, I
think of feeding what?
Specific values into the
machine for x, y, z, and t.
The f machine then
performs on x, y, z, and t
as indicated by f to compute t.
The input of my f
machine in this case
is what I'm going to call a
4-tuple for the time being.
I need four values--
x, y, z, and t.
Order does make a difference.
For example, if I interchange
the x- and the y-coordinate,
those x and y, what I'm doing
is I'm interchanging the x-
and the y-coordinate
of the point in space,
and that in general is going
to change the point in space.
The point, however, is that
in this particular f machine,
notice that my
output is a scalar.
Namely the temperature
is a number,
but the input is a 4-tuple.
x, y, z, and t.
Now the trouble
with using symbolism
like x, y, z, and t, I
guess-- without going
into a long philosophic
discussion--
among other things,
as soon as you
have 27 or more
independent variables,
you run out of letters
of the alphabet.
As a result, it is
quite common for one
to adopt a new notation.
instead of saying
let (x, y, z, t)
be a 4-tuple, what one
usually does is chooses
one symbol-- say x-- and
then uses subscripts.
Namely, a general 4-tuple
would have the form what?
x_1 comma x_2 comma x_3
comma x_4, where x_1,
x_2, x_3 and x_4 are numbers.
An expression like this
is called the 4-tuple.
What is this a
generalization of?
The 4-tuple is a generalization
of the one-dimensional,
two-dimensional, and
three-dimensional arrow,
so to speak, where we
could think of what?
The vector x_1*i as just needing
one number to specify it.
The vector x_1*i plus x_2*j
could've been used to do what?
It could've been abbreviated
by the 2-tuple (x_1, x_2).
And the vector x_1*i plus
x_2*j plus x_3*k could've been
abbreviated by the
3-tuple (x_1, x_2, x_3).
It is conventional
in one-, two-,
or three-dimensional
space to use x, y, and z,
instead of x_1, x_2, and x_3.
But that's just a convention.
I think that it's because we
learnt it that way that we do
it.
In general, I
think the subscript
notation is much nicer, but
in general the idea is what?
Given an ordered array of
n numbers, x_1 up to x_n,
we call that an n-tuple.
And my friend and
colleague John Fitch
mentioned to me
that if n is odd,
like one, three,
five, or seven, then
it's known as an odd-tuple.
Which isn't a very
funny story, that's
why I told you John told
me that particular story.
But the whole idea is
this is an n-tuple.
And the whole idea
again is what?
That an n-tuple makes
sense, even when
n is greater than three.
The whole name of
the game of functions
of several real variables-- in
terms of modern mathematics,
in terms of the
language of n-tuples--
is that a real-valued
function of several--
where by several you mean more
than one-- real variables is
simply a function in
which the input is
an n-tuple and the
output is a number.
OK?
That's what this whole
thing is all about.
And because of that, when we
then abbreviate the n-tuple,
we use x with a bar under it.
Let's call it x-bar.
Rather than x with
the arrow over it,
since arrows may
be inappropriate.
Now what do I mean
by inappropriate?
Well I mean that even
in the case of one, two,
or three dimensions,
you might be thinking
of, say, the radius and
the height of a cylinder,
rather than as an arrow.
And in more than
three dimensions--
for most of us at least--
it's difficult to visualize
what we would mean by an arrow.
So we just use the
bar underneath.
Now again, the major
point is, notice this--
I keep saying the major point.
I guess there's a lot of
major points about this.
Remember that we did not
call arrows "vectors".
We did not call arrows "vectors"
until we defined a structure
on the arrows.
Remember what we did?
We told what it meant for
two arrows to be equal,
we told how we added
two arrows, and we
told how we multiplied
an arrow by a scalar.
In a similar way, we will
not call n-tuples a structure
until we tell how to
equate a pair of n-tuples,
how to add a pair,
and how to multiply
an n-tuple by a number.
By the way, the structure
that we wind up with
is then called an
n-dimensional vector space,
or more concisely, n-space.
And the idea works
like this-- let's
pick a particular value of n.
Lets just call it n.
And let S sub n be the
set of all n-tuples x_1
up to x_n, in other
words, the set of what?
All n-tuples of
numbers x_1 up to x_n.
Let's pick two
particular members
of S sub n, which we'll
call a-bar and b-bar.
Where a-bar is simply
an abbreviation
for the n-tuple a_1 up to a_n,
where the a_1, a_2 up to a_n,
et cetera, are real numbers.
And b-bar is an abbreviation
for the n-tuple b_1,
et cetera, b_n, where
b_1 up through b_n
are also real numbers.
Now again, here's where
structure comes into play.
We have already defined
an n-tuple arithmetic
in terms of arrows
for the case when
n is either one, two, or three.
Based on what happens when
n is one, two, or three,
we invent the
following definitions.
First of all, we
invent the definition
that a-bar equals b-bar means
that the components-- meaning
what?
The individual members of
the n-tuple-- the components
of a-bar are equal to the
components of b-bar, component
by component.
In other words, a_1
is equal to b_1,
a_2 is equal to b_2, et cetera.
All the way up to what?
a_n is equal to b_n.
Now, in other words
again, what we're saying
is that for two
n-tuples to be equal,
by definition, they should be
equal component by component,
and this is
motivated by the fact
that we already know
that we've accepted
this structural definition
for the case of arrows.
Similarly, given two n-tuples
a-bar and b-bar, to add them,
let me define that to be the
n-tuple that I get by adding
component by component.
In other words, to find the
first component of a-bar plus
b-bar, I add the first
component of a-bar
to the first component of b-bar.
Noticing of course,
that this is what?
Be careful here.
This is one number.
a_1 plus b_1 is one number.
a_2 plus b_2 is a number.
a_n plus b_n is a number.
In other words, notice
that by this definition,
the sum of two n-tuples
is again an n-tuple.
And finally, to multiply
a scalar by an n-tuple,
I will agree to
define that definition
to mean that you multiply the
n-tuple component by component
by that particular
scalar, or number.
Notice again that
all I have done here
is I have obtained these
three structural definitions
from the equivalent
situations of arrows.
And since everything that was
true about arrows followed
from these three
basic definitions,
any set of n-tuples that obeys
this particular structure will
also behave like the arrows did.
And that's why we call
it a vector space.
They behave like vectors even
though they can no longer
be viewed as arrows.
And again there are
creative people who
view these things as arrows.
I remember feeling very
intimidated one day
by my undergraduate
professor the first time
I learned vector spaces.
I said, how do you visualize
an n-dimensional vector space?
And in full seriousness,
without batting an eyelash,
he says, "I visualize it
like a porcupine with a bunch
of quills coming out of it."
And I knew that he knew what
was visualizing it like,
but didn't help me one bit.
I'm saying, if you can
visualize this things as arrows,
be my guest.
Feel free to do so.
If you can't, notice that
every one of these definitions
stands on its own two feet.
Subject to the condition that
when n is one, two, or three,
we happen to have a very nice
geometric interpretation.
By the way, I may have
given you the impression
that vector spaces were
invented because of functions
of several variables.
Rather, the impression I
would like to leave you with
is, that in terms of motivating
vector spaces, in terms
of this course, that
was the motivation
that we elected to use.
That the mathematician
talked about vector spaces
in many a different
context from what
we might even dream possible.
In other words, I don't
have to think of temperature
being a function of the four
variables x, y, z, and t.
Let me give you a different
kind of non-trivial example
of a four-space that doesn't
even bring functions into play.
Let's suppose I invent
the abbreviation,
I write the 4-tuple
(a_0, a_1, a_2,
a_3) to denote the cubic
polynomial a_0 plus a_1*x plus
a_2 x squared plus a_3 x cubed.
Notice that I can use these
as a place value system.
The first member tells
me my constant term,
the second member tells
me the coefficient
of x, the third member tells me
the coefficient of x squared,
and the fourth number gives
me the coefficient of x cubed.
Notice that for two polynomials
to be identically equal,
they must be equal, what?
Coefficient by coefficient.
That means what?
Component by component.
How do we add two polynomials?
We add them coefficient
by coefficient.
We add like terms.
In other words, given two
polynomials, we add them what?
Component by component.
We add the two constant terms
together, the two coefficients
of x together, the
two coefficients
of x squared together,
the two coefficients of x
cubed together.
You see?
How do we multiply a
polynomial by a scalar?
We multiply each
term by the scalar.
That, in turn, is equivalent
to multiplying each coefficient
by that scalar.
And that says, in terms
of n-tuple notation,
that we have multiplied each
component by that scalar.
The set of polynomials
of degree n
forms a very nice vector space
in terms of our definition
of a vector space.
Now, of course, the danger
is that one gets the idea
that any set of n-tuples can
be viewed as a vector space.
An n-dimensional vector space.
But this we have to
be careful about.
Remember, it is
not the n-tuples,
it is structure that they obey.
Let me give you sort of a
simple example over here.
Let me consider the
following situation.
First of all, let me just
emphasize a statement;
I just made it, let me
just read it with you.
n-tuples are not
automatically n-spaces.
For example, let me invent the
2-tuple a comma b to represent
the number a plus b.
For example, if I define
the 2-tuple a comma
b to be an abbreviation
for a plus b,
What would 4 comma 5 denote?
Remember the 2-tuple means what?
To get the value of
the 2-tuple is just
the sum of the components.
If I add a and b, in this
case, 4 plus 5 happens to be 9.
How about the 2-tuple 6 comma
3, what value would that have?
That would also
have the value 9.
Therefore numerically,
the 2-tuple 4 comma 5
is equal to the
2-tuple 6 comma 3.
Yet notice that the
first component is not
equal to the first
component here.
In other words 4
is not equal to 6.
Nor is 5 is equal
to 3, but if I were
to choose this
definition of equality,
I could not say that
these 2-tuples form
a two-dimensional vector
space, because it violates
the first definition for
a vector space, namely
the definition of what it means
for two vectors to be equal.
Since we're going to
let most of our material
be covered by the exercises
and the supplementary notes,
and this is just to be an
overview, let's move on now.
Let's assume that we now know
what n-dimensional vector
spaces are like.
We now know that we can view
functions of several variables
as functions that map n-tuples
into numbers, and as a result,
it now makes sense to
talk about things like:
suppose you were given the
n-tuple (x_1, x_2, x_3,
x_3) and suppose that under
f, that n-tuple was mapped
into x_1 cubed plus x_2
plus x_3 squared plus 2*x_4.
For example, if I were to
replace x_1 by 1, x_2 by 3,
x_3 by 1, and x_4 by 2, I would
arrive at the result what?
1 cubed plus 3 plus 1
squared plus 2 times 2,
and I can compute that output.
Now the question that
comes up in calculus is,
can we talk about limits here?
Instead of computing what f of
1 comma 3 comma 1 comma 2 is,
can I compute the limit of this
thing as (x_1, x_2, x_3, x_4)
approaches 1 comma
3 comma 1 comma 2?
I think intuitively it's clear
that since equality means
that you must have equality
component by component,
to say that this
approaches this means
that the first component
here must approach
the first component here.
The second component
here approaches
is the second component
here, et cetera.
In other words, this
could be replaced
by the four separate linear
one-dimensional limit
problems: x_1 approaches 1, x_2
approaches 3, x_3 approaches 1,
and x_4 approaches 2.
And we would then be
tempted to say what?
We will replace x_1 by 1,
x_2 by 3, x_3 by 1, x_4 by 2,
see what happens
to this expression.
And we would then be tempted to
say that this particular limit
was equal to 9.
Now the interesting point is
this-- that traditionally,
this particular problem was
tackled long before anyone
invented vector spaces.
Or at least long
before anybody was
serious about vector spaces.
People did say,
why can't we reduce
the study of
four-dimensional space
to four separate studies
of one-dimensional space?
In other words, let x_1 approach
1, x_2 approach 3, in that case
you're allowing what?
Four separate one-dimensional
limits to be taking place here.
But the insight that
modern math gave
us was that we can now go back
to our traditional definition
of limit.
Remember what was our old
structural definition of limit?
Way back from the
first time we had it.
The limit of f of x as x
approaches a equals L means,
given epsilon greater than
0, we can find delta greater
than zero, such that whenever
the absolute value of x minus a
is greater than 0
but less than delta,
the absolute value of f of x
minus L is less than epsilon
Now here's what
the new math said.
The modern approach
said look, let's just
take our old
structural definition--
the same as before-- and
vectorize everything.
Notice in this situation
that we're dealing with,
the input is where we have
the several variables.
The input is the
n-tuple, the vector, OK?
And the output is the scalar.
So f is a scalar, L is a scalar.
But x and a are
vectors, so every place
I see an x and an a, I have
to put the bar underneath.
And now I read this definition,
and all of a sudden,
as so often has happened
in our course up to now,
I come to something that
I've never seen before.
Namely, as soon
as I look at this.
This made very good sense
when these were arrows.
Namely, we talked
about this earlier
in our course and
one of our lectures.
That to say the two arrows
were near each other
was to say that their
difference was small,
and that in turn said if
the arrows were placed tail
to tail, we could make the
distance between their heads
as small as we wish.
Now the price that we have
to pay for higher dimensions
is that if we have more
dimensions than what
we can draw arrows in, the
problem that we're faced
is that we have not
defined what you
mean by the magnitude of x
minus a, where x and a happen
to be n-tuples.
And here again, we come
back to our structure.
But now for the first time the
structure is not redundant.
Let me tell you
what I mean by that.
In the one-dimensional
case we define
the magnitude of x minus a to
be the square root of x_1 minus
a_1 squared, where
the vector x was
the 1-tuple x_1, and the
vector a was the n-tuple a_1.
In the two-dimensional
case, we said,
OK, let's define the
magnitude of the vector
x minus a to be x_1
minus a_1 squared
plus x_2 minus a_2 squared.
And in the
three-dimensional case,
we said, let's define the
magnitude of x-bar minus a-bar
to be the square root
of x_1 minus a_1 squared
plus x_2 minus a_2 squared
plus x_3 minus a_3 squared.
At that time, I kept saying,
notice that these recipes do
not depend on a picture, that
these are numerical results
that we can compute
without having
to draw a picture at all.
What happened of course was that
in the one-dimensional case,
in the two-dimensional case,
in the three-dimensional case,
it was easier to
visualize the picture.
Now here's where the
real kicker comes in--
and this is the real crucial
point-- structurally,
can't you see what's
happening over here?
Can't you see how I can now
define the absolute value
of the vector x-bar minus a-bar,
even if n is greater than 3,
in such a way that
the definition will
make sense and still mimic
everything that we're doing?
I hope you are a
step ahead of me
on this except for some new
notation I introduced here.
It turns out that in
the modern math book,
one distinguishes between
the absolute value
of a number, and the
magnitude of a vector,
and it is frequently traditional
to introduce a double bar
on each side to represent the
magnitude of the difference
between two vectors,
which I claim behaves
like a distance.
Let me show you
what I mean by that.
Let me define the magnitude
of the n-tuple x-bar
minus the n-tuple a-bar,
written this way, to be
the positive square root
of x_1 minus a_1 squared,
plus et cetera, plus
x_n minus a_n squared.
The thing that I would
like you to notice here
is that since each of these
numbers are non-negative--
see they're squares
of real numbers--
the only way this can be 0--
well the only way that the sum
of squares of non-negative
numbers can be 0
is for each of the
numbers to be 0.
Consequently, the only way the
magnitude of x-bar minus a-bar
can equal 0 is if x_1 equals
a_1, x_2 equals a_2, et cetera,
and x_n equals a_n.
In this vein, notice that
the geometric phrase x-bar
near a-bar still makes sense.
It doesn't make
sense pictorially,
because we can't draw the
arrows if n is greater than 3.
But notice that
what we're saying
is that for x to be
near a, all we're saying
is that the magnitude--
defined this way--
the magnitude of x-bar
minus a-bar is small.
When you're adding
up positive squares,
the only way the sum can be
small is if each of the factors
are small.
But notice what these factors
are, except for the square,
it's the difference between
x_1 and a_1, x_2 and a_2,
et cetera, x_n and a_n.
In other words, to say that
x-bar is near a-bar means
that x_1 is near a_1, x_2
is near a_2, et cetera,
and x_n is near a_n,
which is exactly
the traditional approach.
And in fact, except for the
fact that we can capitalize
on structure,
notice that once we
define the magnitude of
the difference between two
n-tuples-- do you
notice that by the way?
The magnitude of the difference
of two n-tuples is a number.
Notice now if we replace
this fancy phrase--
which we didn't know
the meaning of before,
but which we now know--
by its new definition,
we obtain the traditional
definition of limit.
Namely the limit of f of x_1 up
to x_n, as x_1 approaches a_1,
et cetera, and x_n
approaches a_n,
equals L means that given
epsilon greater than 0,
we can find delta
greater than 0 such
that whenever the square
root of x_1 minus a_1
squared plus et cetera x_n minus
a_n squared is less than delta
but greater than 0, then
the magnitude-- you see,
these numbers
here-- the magnitude
of f of x_1 up to x_n minus
L is less than epsilon.
In, other words,
this definition here
happens to be the
traditional definition.
OK?
But the point is that the
traditional definition
has exactly the same structure
as the modern definition.
And as a result, to make
fun of the traditional math
because it's not as
pretty as the modern math
is the wrong thing to say.
It's like the fellow who
once asked me at a PTA
meeting, "how much is 8 plus
7 in the new mathematics?"
That part hasn't changed.
The beauty of using
the n-tuple notation
was that it allowed us to
use the previous structure
of limits.
So that we can get all of our
theorems, all of our formulas
and what have you, to go
through word for word,
even though the higher the
dimension, the more complex
our computations are.
But structurally, It
essentially boiled down to,
after you've seen
one-dimensional space,
you've seen them all.
That was the big innovation
with the modern approach
to n-dimensional vector spaces.
And to help put this
in proper perspective,
next time I shall
introduce the calculus
of several real variables
in terms of the more
traditional approach.
But again, until
next time, good bye.
Funding for the
publication of this video
was provided by the Gabriella
and Paul Rosenbaum foundation.
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