Hey students, welcome back. This is the last
week of class the last week of this bizarre
coronavirus quarter. So we've got to get through
this week and then we've got finals. We're
starting a new chapter. This is going to be
chapter six. We just finished chapter 5 that was the
material on eigenvalues and now we're moving
on to chapter 6. If you remember back to math
143.
We'll say like this. Don't
use math tool six notations are to remind
you of how that works also recall.
If
you use the column vector.
One in
the column vector. We have
a couple of vectors A bar in, you can define
the dot product.
The dot product
is. And we write it. You be the way you compute
this is you multiply corresponding entries
in atom killer u one v one
plus.
This is the dot product in math 143 you really
focused on vectors that either had two entries
or three entries so r two r three, but you
can define the dot product in general for
vectors are in the same way. Now, the important
thing. The reason that dot products are useful
because it allows us to talk about two things,
the length of a vector and the angle between
two vectors. So, this. So then, I'll say the
link for more sometimes called the norm where
the magnitude. Link
is
in the natural sex book
parallel RS notation to indicate length and
the way you could keep the length is the square
root
of u dot u.
So that's the length and
angle. Yeah.
between C u and v m cosine
of theta equals u
divided by.
So, this formula allows us to talk about the
angle between two vectors. What we're going
to do is generalize this. So my next definition
is going to
take this,
this idea dot product and generalize it so
that we can talk about length, and angles
in other vector spaces
between two polynomials with
the angle between two, two matrices.
So definition.
And,
you know, product that's gonna be the term
that we use for this generalized notion of
dot products on the inner product on some
vector space V is a function that
associates to
pair.
u and v in V
real number,
which we will right now
instead of putting dots between the two vectors,
the notation is angle bracket, u comma V,
so this is going to be a number.
And there's gonna be four properties on the
right these properties
first property is as commutativity.
Also, a distributive property.
The inner product
of u plus v with w is
equal to the inner product.
I'm also work nicely
with scalars so
Kenny, you
k You, me, pay
more property
inner product
of you with itself revenue equals zero, and
we already have zero, so inner product of
new is
equal to zero if and only
the inner product then it's just this function,
each time you take a couple of vectors in
associated. It's out. Also, a real number.
feature
vector face. Better.
Can your product.
Right. And
so that's we're going to be studying for the
next few days, or your product tastes.
The first obvious example
are in
the inner product of u v
equals. So,
you is one example you can
check with satisfies those four properties.
So the dot product is space. Now, here's what
we're going to do if v is an inner product.
We've got some inner product that we'll denote
with angle brackets
and a U and V, V. We define the norm or the
length. So we define
the norm.
length, I tend to use
these two words interchangeably,
the norm or length.
You
is. And we'll use double parallel wars around
you, and how do we compute it.
The inner product of you with itself,
things where
we also define the distance so going on in
distance
between you.
Okay, so now what I want to do. Oh look at
various record spaces and show familiar vector
spaces
that we've been working with all have inner
products
so spaces.
First one
was called the weighted
products
are so we've seen that are in has the enterprise
read the dot product but
that's not the only one. There's also this
one called a way to get in.
So what I want to do is pick some positive
numbers so x
column.
You want to fix positive
numbers.
One
and one is the sign.
As a callback for someone are in, so you want
people to
find
the inner product of u and
v so this is going to have an alternate formulation,
and it'll go like this, W one times u one
v one plus two,
plus n plus n.
Turns out that this formula right here defines
an inner product space or finding a product
satisfies the four properties that we discussed.
Let's just do an example here.
So, in
case that was working for free so let's say
n equals three. I need out three positive
number. So let's take w one equal to three
equal to one, u three equal to two.
Now, yes.
Let's say u is the column vector, 102,
and
v is the column vector. Negative 31. Right
now, when I compute the inner product a weighted
inner product
center. So then
enter product
is going to be three times one.
They are free three
plus one times zero times one plus two
times two times four,
so it's very close to the dot product what
changes, though, is that we have these weights
out in front. It's kind of like a dumbbell.
So when we get here, negative nine
plus 16. So that is seven.
So there's an example of sort of alternate
your products.
Okay, so
I'm gonna do more examples but first this
definition so if see
his inner.
This fear
is that of all
U and V,
such that
the norm or the length
is equal to one so it's the set of all vectors
in your vector space in your inner product
space
that length one
in this special case so in
the dimension.
V equals two, instead of saying the unit sphere
we say unit circle or
sphere. We use the obvious word unit circle
and
let's see let's let's look at an example here.
So, in red.
We use
enterprise
view dot v
is equal to u dot v.
So, if our inner 
product
is not cross
circle right we're working in dimension to
create a circle looks exactly.
You think it really looks like a circle.
Type set of factors are two when you're working
with a dot product right board is exactly
equal to those that lie on or look towards
like a circle. Again,
or two.
So now what I want to do is
I want to change the inner product now I want
to work with a weighted inner product. Don't
say else to our products, where w one 
equals
four.
To me holes.
So what I like to do is see what a circle
looks like now.
Yes.
He holds the vector
on YouTube.
You 
u equals one.
C so that's going to be four one squared plus
night.
Choose work.
Radiant so for us he wants. Yeah.
And then I could rewrite this.
This is if and only if
you weren't squared over
four and a half squared
plus two squared
over three squared equals one and rewriting
this way so that it becomes apparent that
while we're talking about
the lips.
If I think of the vertical axis as the you
two axis, y axis, and
you are the horizontal axis if
you want axis. What we have
our unit circle. This rate intercom looks
different than standard enterprise so this
would be a.
This would be three.
So, the unit circle is definitely dependent
on the way you compute link. If you change
the way you compete wave. Then what you end
up with
is
one thing I should point out
the dot product is also referred to as
the dot product 
on
is also you
as
clean
as you hear clean enterprise products.
Excel see some examples of inner products
and vector spaces, other than. So, we've got
some familiar ones. Let's look at matrices
first, so I'm given.
Let's say matrices
A and B. in.
So we've
got a couple of n by n matrices. Now we're
going to define
the inner product
of the matrices, a way
that's going to be equal to
trace.
A transpose.
Oh, we haven't seen trace since very beginning,
remember, the trace is the sum over guy
all entries.
So, it turns out that this defines an inner
product which gives us a way to talk about
the length of a matrix, it tells it gives
us a way an angle where we'll see that we
need to buy an angle also. So it gives us
a way to think about length and distance between
matrices.
So for example.
In, two, three.
Let's look at a
matrix, 102213.
And B
is matrix.
021
minus 111.
So, if I compute
compute a transpose.
So that's going 
to be
times.
Okay, so this is going to 
be equal to.
224321 negative one, negative 134131, and
six.
So, the inner products.
Okay with B is going to be the trace of this
matrix right here, and this
is like.
So we just talked about the
inner products on matrices. Now
let's look at what's called the standard inner
product
on on
the space of polynomials
so standard.
And it goes like this. So, given.
ordinals
equal to a zero plus a one
x to the n is
equal to zero. dot.
We've got a couple of polynomials. To find
what's called the standard inner product.
So, the inner product.
p of x cube x is going to be defined kind
of an obvious way. What I'm going to do is
multiply the corresponding coefficients so
it's gonna be a zero,
and feasor.
A one,
you check with the satisfies the properties.
So for example,
let's say
here is.
plus two x minus x squared,
q of x
is x plus three. So here we're in
peace, he's up to.
Between
these two as well as
me three times zero
plus two times one plus minus one times three,
answer that gives us. Okay. Well, one more
important in our province. And I should point
out. I can't show every single possible example
of an inner product right so in the book
you will see fewer examples of enterprise
including
another inner product.
But let's just look at this one here so
on.
vector space consisting of all continuous
functions from A to Z we haven't talked about
function spaces a whole lot
in this class, but they become
important especially when you're working with
differential equations. So let's define the
inner product of two functions to be the integral
from a to b of f of x cube x. Yes. It turns
out this formula right here, y functions and
integrate them over the interval defines an
inner product. And so for example. So, so
on.
t zero to two pi.
If I wanted to compute the integral
between the function sine x
function, sine two x.
That would be equal to the integral from zero
to pi
sine of x, sine two x. Yes.
Keep this game going you need to remember
trig identity.
So
sine of two x is the same as two sine x.
Back to
142, so this is equal to 20225.
This will be a sine squared x
times cosine x.
Yes, you can integrate this by substitution,
where you let u equal x u equals cosine x,
x, so this becomes
integral of e to the integral from zero to
zero. u squared u is equal to zero. So the
inner product of these two functions sine
x and sine two x right turns out to be zero.
We'll talk about angles. Next time, but one
thing to
remember if you think back to dot products.
What did it mean, it does not equal to zero.
It meant that the two vectors needed a right
angle. In generalized that idea once I'm talking
about the angle between two vectors. So, the
orthogonal. intercrop. So here's one thing
that's hard to picture of when this computation
right here just showed
that these two functions sine of x and sine
two
x
array. Okay.
Transcribed by https://otter.ai
