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5DUQ3-Y_gX4
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ALAN EDELMAN: I call it the
way for big boys and big girls.
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5DUQ3-Y_gX4
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STEVEN G. JOHNSON:
Yes, for big kids.
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5DUQ3-Y_gX4
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ALAN EDELMAN: For grown-ups.
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5DUQ3-Y_gX4
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STEVEN G. JOHNSON:
The big-kid way.
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5DUQ3-Y_gX4
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ALAN EDELMAN: Kid way.
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5DUQ3-Y_gX4
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STEVEN G. JOHNSON: OK.
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5DUQ3-Y_gX4
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So df, let's just do it slowly.
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5DUQ3-Y_gX4
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So what we want to do
is we want to take f.
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5DUQ3-Y_gX4
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We're going to take--
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5DUQ3-Y_gX4
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I'm going to draw my
vector symbols here.
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5DUQ3-Y_gX4
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I guess I'll put them here.
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5DUQ3-Y_gX4
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But I get tired of
writing them all the time.
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5DUQ3-Y_gX4
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All my x's, and therefore
my dx's, are vectors.
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5DUQ3-Y_gX4
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So think of it as an
arbitrary small change
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5DUQ3-Y_gX4
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in an arbitrary direction.
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5DUQ3-Y_gX4
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We want it to be able to
handle anything like that.
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5DUQ3-Y_gX4
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ALAN EDELMAN: And
in case it wasn't
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5DUQ3-Y_gX4
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already obvious to everybody,
what is the output of f?
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5DUQ3-Y_gX4
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Is it a scalar, a
vector, a matrix?
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5DUQ3-Y_gX4
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It's a scalar, exactly.
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5DUQ3-Y_gX4
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Just wanted to make sure
everybody realized--
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5DUQ3-Y_gX4
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STEVEN G. JOHNSON: Sorry, yes.
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5DUQ3-Y_gX4
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ALAN EDELMAN: --that this is
a scalar function of a vector.
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5DUQ3-Y_gX4
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STEVEN G. JOHNSON: Yeah.
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5DUQ3-Y_gX4
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You could also write this
as x dot product with ax.
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5DUQ3-Y_gX4
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That's the same thing.
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5DUQ3-Y_gX4
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OK.
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5DUQ3-Y_gX4
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So I'm just going
to do this out.
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5DUQ3-Y_gX4
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I think it's still a
little bit laboriously,
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5DUQ3-Y_gX4
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but we'll have a better rule
for-- we'll do the product
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5DUQ3-Y_gX4
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rule in a minute.
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5DUQ3-Y_gX4
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But let's do it without
the benefit of that.
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5DUQ3-Y_gX4
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ALAN EDELMAN: Because
you're effectively
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5DUQ3-Y_gX4
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deriving the product rule
in what's about to come.
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5DUQ3-Y_gX4
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STEVEN G. JOHNSON: Exactly, yes.
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5DUQ3-Y_gX4
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So what do we do?
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5DUQ3-Y_gX4
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So I'm going to plug--
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5DUQ3-Y_gX4
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I'm going to take
f of x plus dx.
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5DUQ3-Y_gX4
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I'm going to subtract fx.
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5DUQ3-Y_gX4
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And I'm going to drop--
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5DUQ3-Y_gX4
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because it's d's, I'm
going to drop anything
|
5DUQ3-Y_gX4
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that looks like a d squared,
a dx squared, something
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5DUQ3-Y_gX4
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that goes to 0 faster than dx.
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5DUQ3-Y_gX4
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So what's f of x plus dx?
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5DUQ3-Y_gX4
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Well, I take x.
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5DUQ3-Y_gX4
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I add dx, and I plug it into f.
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5DUQ3-Y_gX4
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So there's a transpose there.
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5DUQ3-Y_gX4
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There's an A. There's
an x plus dx there.
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5DUQ3-Y_gX4
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And then I subtract
x transpose Ax.
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5DUQ3-Y_gX4
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And then I can just
multiply everything out.
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5DUQ3-Y_gX4
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Just think of dx as a
really small vector.
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5DUQ3-Y_gX4
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And just use your ordinary
rules from linear algebra.
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5DUQ3-Y_gX4
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I can just use the
distributed whatever it's
|
5DUQ3-Y_gX4
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called, the distributive rule.
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5DUQ3-Y_gX4
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I just have to make sure I don't
change the orders of anything
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5DUQ3-Y_gX4
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since these are
vectors and matrices.
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5DUQ3-Y_gX4
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So I can multiply.
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5DUQ3-Y_gX4
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There's this term times
this term times this term.
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5DUQ3-Y_gX4
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That's the first term.
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5DUQ3-Y_gX4
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So there's an x transpose Ax.
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5DUQ3-Y_gX4
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There's also this term times
this term times this term.
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5DUQ3-Y_gX4
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So that's a dx transpose Ax.
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5DUQ3-Y_gX4
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So dx is just a little vector.
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5DUQ3-Y_gX4
|
It's perfectly fine
to transpose it.
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5DUQ3-Y_gX4
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And then I also have
this term times this term
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5DUQ3-Y_gX4
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times this term.
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5DUQ3-Y_gX4
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So that's x transpose Adx.
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5DUQ3-Y_gX4
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And then I have this term times
this term times this term.
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5DUQ3-Y_gX4
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But that has a dx squared.
|
5DUQ3-Y_gX4
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And so that I'm going to just--
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5DUQ3-Y_gX4
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let's see.
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5DUQ3-Y_gX4
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Well, I actually am going
to [INAUDIBLE] here.
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5DUQ3-Y_gX4
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So that term is your
dx transpose Adx.
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5DUQ3-Y_gX4
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This term is gone.
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5DUQ3-Y_gX4
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This is high order.
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5DUQ3-Y_gX4
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So in the limit as dx
gets smaller and smaller
|
5DUQ3-Y_gX4
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and smaller, this
term is negligible
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5DUQ3-Y_gX4
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compared to these terms.
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5DUQ3-Y_gX4
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And then I still have this term
over here, can't forget that.
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5DUQ3-Y_gX4
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Otherwise, these terms are
negligible compared to this.
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5DUQ3-Y_gX4
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But it's OK because I'm
going to subtract that.
|
5DUQ3-Y_gX4
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And that term cancels.
|
5DUQ3-Y_gX4
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And what's left is
these two terms.
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5DUQ3-Y_gX4
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And this is a perfectly
good linear operation on dx,
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5DUQ3-Y_gX4
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but it's not written
in a very nice form.
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5DUQ3-Y_gX4
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So the trick is to--
since these are numbers,
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5DUQ3-Y_gX4
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I can transpose them.
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5DUQ3-Y_gX4
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And this is something I
feel like in 18.06 people
|
5DUQ3-Y_gX4
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get very confused by,
that normally you're
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5DUQ3-Y_gX4
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not allowed to change
the order of anything.
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5DUQ3-Y_gX4
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Normally, a matrix is not
equal to its transpose.
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5DUQ3-Y_gX4
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But a number is always
equal to its transpose.
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5DUQ3-Y_gX4
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So this is since--
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5DUQ3-Y_gX4
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let me do a note over here.
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5DUQ3-Y_gX4
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Since dx transpose
Ax is a scalar,
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5DUQ3-Y_gX4
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a scalar is always
equal to its transpose.
|
5DUQ3-Y_gX4
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So then we can take
this and set it
|
5DUQ3-Y_gX4
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equal to its own transpose,
which is the same thing as x
|
5DUQ3-Y_gX4
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transpose A transpose dx.
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5DUQ3-Y_gX4
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And again, make sure
you understand that.
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