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ZK3O402wf1c
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How could it go wrong
that out of these --
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out of three columns and
all their combinations --
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when would I not be able
to produce some b off here?
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When could it go wrong?
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ZK3O402wf1c
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Do you see that
the combinations --
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ZK3O402wf1c
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let me say when it goes wrong.
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If these three columns
all lie in the same plane,
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ZK3O402wf1c
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then their combinations
will lie in that same plane.
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ZK3O402wf1c
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So then we're in trouble.
|
ZK3O402wf1c
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If the three columns
of my matrix --
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ZK3O402wf1c
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if those three vectors happen
to lie in the same plane --
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ZK3O402wf1c
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for example, if
column three is just
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the sum of column one and column
two, I would be in trouble.
|
ZK3O402wf1c
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That would be a matrix A
where the answer would be no,
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ZK3O402wf1c
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because the combinations --
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ZK3O402wf1c
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if column three is in the same
plane as column one and two,
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ZK3O402wf1c
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I don't get anything
new from that.
|
ZK3O402wf1c
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All the combinations are in the
plane and only right-hand sides
|
ZK3O402wf1c
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b that I could get would
be the ones in that plane.
|
ZK3O402wf1c
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So I could solve it for
some right-hand sides, when
|
ZK3O402wf1c
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b is in the plane, but
most right-hand sides
|
ZK3O402wf1c
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would be out of the
plane and unreachable.
|
ZK3O402wf1c
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So that would be
a singular case.
|
ZK3O402wf1c
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The matrix would
be not invertible.
|
ZK3O402wf1c
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There would not be a
solution for every b.
|
ZK3O402wf1c
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The answer would
become no for that.
|
ZK3O402wf1c
|
Okay.
|
ZK3O402wf1c
|
I don't know --
|
ZK3O402wf1c
|
shall we take just a
little shot at thinking
|
ZK3O402wf1c
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about nine dimensions?
|
ZK3O402wf1c
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Imagine that we have vectors
with nine components.
|
ZK3O402wf1c
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Well, it's going to be
hard to visualize those.
|
ZK3O402wf1c
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I don't pretend to do it.
|
ZK3O402wf1c
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But somehow, pretend you do.
|
ZK3O402wf1c
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Pretend we have -- if this
was nine equations and nine
|
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unknowns, then we would
have nine columns,
|
ZK3O402wf1c
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and each one would be a vector
in nine-dimensional space
|
ZK3O402wf1c
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and we would be looking at
their linear combinations.
|
ZK3O402wf1c
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So we would be having
the linear combinations
|
ZK3O402wf1c
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of nine vectors in
nine-dimensional space,
|
ZK3O402wf1c
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and we would be trying to
find the combination that hit
|
ZK3O402wf1c
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the correct right-hand side b.
|
ZK3O402wf1c
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And we might also ask the
question can we always do it?
|
ZK3O402wf1c
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Can we get every
right-hand side b?
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ZK3O402wf1c
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And certainly it will depend
on those nine columns.
|
ZK3O402wf1c
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Sometimes the answer
will be yes --
|
ZK3O402wf1c
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if I picked a random matrix,
it would be yes, actually.
|
ZK3O402wf1c
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If I used MatLab and just used
the random command, picked
|
ZK3O402wf1c
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out a nine by nine matrix,
I guarantee it would be
|
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good.
|
ZK3O402wf1c
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It would be
non-singular, it would
|
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be invertible, all beautiful.
|
ZK3O402wf1c
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But if I choose those columns
so that they're not independent,
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ZK3O402wf1c
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so that the ninth column is
the same as the eighth column,
|
ZK3O402wf1c
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then it contributes
nothing new and there
|
ZK3O402wf1c
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would be right-hand sides
b that I couldn't get.
|
ZK3O402wf1c
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Can you sort of think
about nine vectors
|
ZK3O402wf1c
|
in nine-dimensional space
an take their combinations?
|
ZK3O402wf1c
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That's really the
central thought --
|
ZK3O402wf1c
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that you get kind of used
to in linear algebra.
|
ZK3O402wf1c
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Even though you can't
really visualize it,
|
ZK3O402wf1c
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you sort of think you
can after a while.
|
ZK3O402wf1c
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Those nine columns and
all their combinations
|
ZK3O402wf1c
|
may very well fill out the
whole nine-dimensional space.
|
ZK3O402wf1c
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But if the ninth column happened
to be the same as the eighth
|
ZK3O402wf1c
|
column and gave nothing new,
then probably what it would
|
ZK3O402wf1c
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fill out would be --
|
ZK3O402wf1c
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I hesitate even to say this --
it would be a sort of a plane
|
ZK3O402wf1c
|
--
|
ZK3O402wf1c
|
an eight dimensional plane
inside nine-dimensional space.
|
ZK3O402wf1c
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And it's those eight
dimensional planes
|
ZK3O402wf1c
|
inside nine-dimensional
space that we
|
ZK3O402wf1c
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have to work with eventually.
|
ZK3O402wf1c
|
For now, let's stay with a nice
case where the matrices work,
|
ZK3O402wf1c
|
we can get every
right-hand side b and here
|
ZK3O402wf1c
|
we see how to do
it with columns.
|
ZK3O402wf1c
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Okay.
|
ZK3O402wf1c
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There was one step
which I realized
|
ZK3O402wf1c
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I was saying in words that I
now want to write in letters.
|
ZK3O402wf1c
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Because I'm coming back to the
matrix form of the equation,
|
ZK3O402wf1c
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so let me write it here.
|
ZK3O402wf1c
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The matrix form of my
equation, of my system
|
ZK3O402wf1c
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is some matrix A
times some vector x
|
ZK3O402wf1c
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equals some right-hand side b.
|
ZK3O402wf1c
|
Okay.
|
ZK3O402wf1c
|
So this is a multiplication.
|
ZK3O402wf1c
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A times x.
|
ZK3O402wf1c
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Matrix times vector,
and I just want to say
|
ZK3O402wf1c
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how do you multiply
a matrix by a vector?
|
ZK3O402wf1c
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Okay, so I'm just going
to create a matrix --
|
ZK3O402wf1c
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let me take two
five one three --
|
ZK3O402wf1c
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and let me take a vector
x to be, say, 1and 2.
|
ZK3O402wf1c
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How do I multiply a
matrix by a vector?
|
ZK3O402wf1c
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But just think a little
bit about matrix notation
|
ZK3O402wf1c
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and how to do that
in multiplication.
|
ZK3O402wf1c
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So let me say how I multiply
a matrix by a vector.
|
ZK3O402wf1c
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Actually, there are
two ways to do it.
|
ZK3O402wf1c
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Let me tell you my favorite way.
|
ZK3O402wf1c
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It's columns again.
|
ZK3O402wf1c
|
It's a column at a time.
|
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