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Okay. So we are, let's move on to the
third video of this lecture, which is on

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Tensor products. Let me just say a word
before we go on. Which is, first, for many

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of you, this might be too much all at
once. So you know, take it in whatever

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doses you, you, you can digest it and
let's see. So, some of this stuff, you

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know, some of the things that I'm speaking
about, of course. It's just a

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formalization of stuff you've already seen
in the, in the previous lectures.

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Especially these, tensor products. You
know, we've been implicitly using them for

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the last couple of weeks. And, hopefully
this only, you know, this only help, helps

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clarify to you what the, what the
underlying math behind it, behind it all

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is. Okay, so. So, with that, with that
small warning let's, let's just go on and

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let's talk about, tensor products. So,
let's, let's go back and remember, what a

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qubit was. So, so you remember we, you
know, our picture of it, was, was like

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this. We have a hydrogen atom. Its
electron, we think of it as being either

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in the ground state, which we think of as
a, as representing zero, and an excited

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state, which we think of as a, as
representing one. And now, of course, the

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state of this, this qubit, is, is a unit
vector in. So it's, it's some unit vector,

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phi in a two dimensional complex vector
space, which we call a Hilbert space.

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Okay. We call it a Hilbert space mainly
because it's a, it's a complex vector

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space with a defined inner product. And
then. You know this is additional property

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that we want from Hilbert's spaces when,
when they are, you know which is more

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complicated, when they are infinite, and
infinite dimensional Hilbert spaces so

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when we talked about continuous quantum
states. But, in this course, we are not

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going to deal with all those subtleties
about their completeness, you know limits

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and so on. So formerly we'll only think
about finite eventual Hilbert spaces. And

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then when we talk about continuous quantum
states, we'll just do it intuitively.

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Okay, so, so , back here, okay, so we
have, we have a cubit, it's a unit vector

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in this two dimensional complex vector
space, which we're calling a Hilbert

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space, and. And now, let's say we have
another such electron in another hydrogen

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atom. And so, of course, it's state also
is a unit vector in a, in a two

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dimensional Hilbert space. Okay, but, but
now, of course. In general, we know that

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we cannot really talk about if we are
given the two hydrogen atoms together, we

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have, we are given these two qubits
together, we cannot in general talk about

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the state of the first qubit and the state
of the second qubit separately because of

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the phenomenon of entanglement. So, in
general, we cannot talk about, any more

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about the state of the first qubit and the
state of the second qubit, but we still

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have these two Hilbert spaces associated
with the first qubit and the second qubit.

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So now, what happens when we bring these
two qubits together? Okay, so what you

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already saw is that. When we think of this
as a composite system, there is actually a

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four dimensional complex vector space
associated with it. And the basis for this

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state is all the classical possibilities
which is 0-0, 0-1, 1-0, and 1-1. But

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still, you can ask what was the operation
by which we, we took two such complex

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vector spaces, two dimensional complex
vector spaces. And we glued them together

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to get a four dimensional complex vector
space. What is this operation? So if you

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call this Hilbert space H1, and we call
this Hilbert space H2, then what's the

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operation that they perform on these two
Hilbert spaces to get this new Hilbert

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space H, which is a four dimensional
complex vector space. So, this operation

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is called a tensor product. And when we
perform this tensor product, what we do

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is, we look, we take a basis vector from
the first Hilbert space, like zero and a

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basis vector from the second vector space,
like zero. And we take a tensor product

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between them. Okay. And, well it might be.
You know, just to, just for clarity, it

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might be, it might be a little, easier for
you t o understand if I write them a

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little closer together. So, I take. Okay.
So, let's, let's, let's erase all this,

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and let's call this, let's call this, this
one, H1. H2, and we've taken a tensor

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product between them. So, we are taking a
basis vector from the first Hilbert space,

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one from the second Hilbert space, and we
take the tensor product of them. And this

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tensor product we also write as so, when
you write 0x0 we also you know this is

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what we have been calling 00. Right? And
sometime we will also write just as cat

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zero, cat zero. So, this is what we mean
by saying well the, the first electron was

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in the state zero and the second electron
was in the state zero. Similarly, if you

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take the tensor product of these two, we,
we can also abuse notation and write it

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like, like this or like that, and so on.
For the, for the, for the, for each of the

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four cases. Okay and then, okay so this is
how we glue these two Hilbert spaces

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together and got a new Hilbert space, new
complex vector space, which is now four

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dimensional. Okay, we can also ask, you
know, of course we can, we can glue

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together any vector here and any vector
here so if you have, if you have a state

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phi one in the first Hilbert space, and
phi two in the second Hilbert space, we

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could, we could glue them together by
putting this, this symbol in between them.

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And this is what, you know, so for
example, if we had if we had the state one

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over square root to zero plus one over
square root to one, and the first overt

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space and one over square root to zero
minus one over square root to one in the

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second overt space. And if it took a
product of these, a tensor product, this

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is what we were, we were writing
informally just by writing a product. And

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now, when we take tensor products it's
distributive, as you know, exactly the

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rules that we followed. And we'd write
down series. One half zero times tensor to

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zero minus one half zero tensor to one
plus one half one tensor of zero minus one

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half, one tensor of one. Okay. That's what
the that's what the tensor product of two

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states would look like. We could also a
sk, well what happens if you, if you look

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at two set states? Phi one tensor and phi
two and phi one tensor at phi two. What's

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the inner product between them? And the
answer is, it's just the inner product

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between phi one and phi one. So, whatever
the states are in the first, in the first

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Hilbert space times the inner product
between phi two and phi two. Okay, finally

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lets look at the more general case. So,
now suppose we have a, we have a K-state

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particle. A K-state, sorry a K-state
quantum system. So, if the first Hilbert

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space is a K-dimensional Hilbert space.
And let's say, let's, let's say the second

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Hilbert space is, is made up of an L-state
quantum system. So H2 is an L-dimensional

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complex vector space. So this, has a basis
zero through K - one, let's say. And let's

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call these basis vectors, say, zero
through, L - one. Okay, so now what

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happens when we glue these two together,
when we put them together and call this

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one big system? So, we get some Hilbert
space H. And H is a tensor product of H1

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and H2. Okay, what's, what's another way
of saying it? Well another, you know the

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intuitive way of saying it is look, you
had a quantum system here, the first one,

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where you could be in one of these key
distinguishable states, zero through K -

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one, or in any super-position of these. In
each two you can be in one of L

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distinguished will states. Why need super
positions of those? And so when you put

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them together, what are the
distinguishable states? Well, you could be

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in 00, 01, 02 through 0L - one, or ten,
eleven through 1L - one. So, they are

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exactly K times L different
distinguishable states. So H, its a K

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times L dimensional space. And again. You
know, the, the, the basis vectors in H are

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going to be tensor, tensor products of
basis vectors in H1 and H2. So they're

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going to be of the form zero tensor zero.
Zero tensor one. Zero tensor L-1. One

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tensor zero one tensor L-1, all the way to
K-1 tensor zero. K-1 tensor L-1. Okay, so

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there are exactly K times L of these. And
of course, your state can be any linear

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combination of these. Again, remember by
abuse of notation, w e, we, we can, we are

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going to denote the state, also by. By
this or also by. By that. Okay. So now,

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there's something to think about here.
There's something very interesting going

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on here. Okay, what, what's this
interesting thing? So, if you were to

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write a state of your first system, how
many parameters would you need for that?

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Well since it's a K-dimensional system,
you need. K. Complex numbers to describe

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your state in of, of the first system. How
many, how many parameters would you need

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if you were just to put, describe a state
in your second system? Well, you need

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exactly L-complex numbers to describe a
state in your second system. But now, if

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you put them together, if you call your
system the union of, of your first and

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second quant, quantum system. Then you
need K L complex numbers. Okay, so it's

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worth thinking about this intuitively.
What would have happened if this was a

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classical system? Let's say that, You
know, you had some system where you needed

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a million parameters to describe, describe
your system. So, for example, what's,

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what's, a, what's a system where you need
a million parameters to describe it? Well,

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let's say it's the memory in your
computer. So, let's say that you got a,

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got a mem, you know, you know, you, you,
you bought a mega, megabit, megabyte of,

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memory. So, ten^6. You need ten^6 numbers
to describe what the state of that memory

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is. And now you are running short of
memory. So, you go out and purchase

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another, you know, some more rams, so its,
lets say you purchase other ten^6

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megabytes. How much memory do you have?
Well, you have ten^6 + ten^6 which is two

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ten^6. Alright? They add up, so the number
of the parameters you need to describe the

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state of the existence once you have, once
you have, one you add these new memories

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in, it's just the sum of the parameters
you need for the first one and the

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parameters you need for the second one.
What happens if this was a complex system,

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sorry, quantum system? So, you needed
ten^6 parameter for your first system,

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your first quantum memory , ten^6 for the
second one. Right? If you use them

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individually. But if you put them
together, and call the whole thing one big

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system. Now you need ten^6 ten^6. So,
instead of a megabyte, you now end up with

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a terabyte of memory. This sounds
ridiculous. But this is, this is really,

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you know, this is the basic fact about
quantum mechanics or quantum systems which

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make them so strange and therefore also so
useful from a computational viewpoint.

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Okay, one last thing to help you think
about this. So, how could it be that you

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needed only K-complex numbers to describe
a state in H1. L to describe a state in

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H2. But now, if you put, put the two
systems together you need K L complex

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numbers. How could this be? What, what's
going on? Well the answer is,

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entanglement. Okay, so, so why is the
answer entanglement. So. So you, you had

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H1 but you needed K-complex numbers, you
had H2 but you needed L-complex numbers.

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So if you had a state, phi one, you need
only K-parameters to describe phi one. A

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state of H, of, of the first quan, quantum
system. If you have phi two, the state of

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the second quantum system, you only need
L-parameters, L-complex numbers to

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describe it. If your have state of the two
systems is of the form phi one tensor phi

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two, you only need K + L parameters to
describe it. But remember, the state of a

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composite system does not need to be a
product state. In general, it's going to

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be an entangled state. And these states
cannot be described using K-parameters on

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the, on the first space and L-parameters
on the second space. And that's the reason

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why, in general most of the states of
this, of this composite system are going

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to be entangled. And this is why we need K
L parameters to describe a general state

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of this composite, quantum system, which
we describe by the Hilbert space H, which

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is H1 tensor H2. Okay? So, how much of
this should you understand? Well, you

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know, it depends upon, it depends upon how
deeply you want to get into the subject.

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But hopefully this gives you, some sense
of the, you know, of sol id ground, in

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terms of, what are all these symbols that
we've been using so far? What do all these

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operations we've been using so far mean?
So use it, you know, use this, this

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lecture. I, I, I guess What I should say
is try to digest as much of this as you

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find comfortable, and you know, I'll try
to be as gentle as possible going forward,

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and use, use this notation as sparingly as
possible. Okay.