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ZK3O402wf1c
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It's because we already know
that that's the right x and y,
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ZK3O402wf1c
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so why not take the correct
combination here and see it
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ZK3O402wf1c
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happen?
|
ZK3O402wf1c
|
Okay, so how do I picture
this linear combination?
|
ZK3O402wf1c
|
So I start with this vector
that's already here --
|
ZK3O402wf1c
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so that's one of column
one, that's one times column
|
ZK3O402wf1c
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one, right there.
|
ZK3O402wf1c
|
And now I want to add on --
so I'm going to hook the next
|
ZK3O402wf1c
|
vector onto the front of the
arrow will start the next
|
ZK3O402wf1c
|
vector and it will go this way.
|
ZK3O402wf1c
|
So let's see, can I do it right?
|
ZK3O402wf1c
|
If I added on one
of these vectors,
|
ZK3O402wf1c
|
it would go left one and up two,
so we'd go left one and up two,
|
ZK3O402wf1c
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so it would probably
get us to there.
|
ZK3O402wf1c
|
Maybe I'll do dotted
line for that.
|
ZK3O402wf1c
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Okay?
|
ZK3O402wf1c
|
That's one of column
two tucked onto the end,
|
ZK3O402wf1c
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but I wanted to tuck
on two of column two.
|
ZK3O402wf1c
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So that -- the second one --
we'll go up left one and up two
|
ZK3O402wf1c
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also.
|
ZK3O402wf1c
|
It'll probably end there.
|
ZK3O402wf1c
|
And there's another one.
|
ZK3O402wf1c
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So what I've put in here
is two of column two.
|
ZK3O402wf1c
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Added on.
|
ZK3O402wf1c
|
And where did I end up?
|
ZK3O402wf1c
|
What are the coordinates
of this result?
|
ZK3O402wf1c
|
What do I get when I take
one of this plus two of that?
|
ZK3O402wf1c
|
I do get that, of course.
|
ZK3O402wf1c
|
There it is, x is zero,
y is three, that's b.
|
ZK3O402wf1c
|
That's the answer we wanted.
|
ZK3O402wf1c
|
And how do I do it?
|
ZK3O402wf1c
|
You see I do it just
like the first component.
|
ZK3O402wf1c
|
I have a two and a minus
two that produces a zero,
|
ZK3O402wf1c
|
and in the second component I
have a minus one and a four,
|
ZK3O402wf1c
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they combine to give the three.
|
ZK3O402wf1c
|
But look at this picture.
|
ZK3O402wf1c
|
So here's our key picture.
|
ZK3O402wf1c
|
I combine this column and
this column to get this guy.
|
ZK3O402wf1c
|
That was the b.
|
ZK3O402wf1c
|
That's the zero three.
|
ZK3O402wf1c
|
Okay.
|
ZK3O402wf1c
|
So that idea of linear
combination is crucial,
|
ZK3O402wf1c
|
and also --
|
ZK3O402wf1c
|
do we want to think
about this question?
|
ZK3O402wf1c
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Sure, why not.
|
ZK3O402wf1c
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What are all the combinations?
|
ZK3O402wf1c
|
If I took -- can I
go back to xs and ys?
|
ZK3O402wf1c
|
This is a question for really --
|
ZK3O402wf1c
|
it's going to come
up over and over,
|
ZK3O402wf1c
|
but why don't we
see it once now?
|
ZK3O402wf1c
|
If I took all the xs and all
the ys, all the combinations,
|
ZK3O402wf1c
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what would be all the results?
|
ZK3O402wf1c
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And, actually, the
result would be
|
ZK3O402wf1c
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that I could get any
right-hand side at all.
|
ZK3O402wf1c
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The combinations
of this and this
|
ZK3O402wf1c
|
would fill the whole plane.
|
ZK3O402wf1c
|
You can tuck that away.
|
ZK3O402wf1c
|
We'll, explore it further.
|
ZK3O402wf1c
|
But this idea of what linear
combination gives b and what do
|
ZK3O402wf1c
|
all the linear
combinations give,
|
ZK3O402wf1c
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what are all the possible,
achievable right-hand sides be
|
ZK3O402wf1c
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-- that's going to be basic.
|
ZK3O402wf1c
|
Okay.
|
ZK3O402wf1c
|
Can I move to three
equations and three unknowns?
|
ZK3O402wf1c
|
Because it's easy to
picture the two by two case.
|
ZK3O402wf1c
|
Let me do a three
by three example.
|
ZK3O402wf1c
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Okay, I'll sort of
start it the same way,
|
ZK3O402wf1c
|
say maybe 2x-y and maybe I'll
take no zs as a zero and maybe
|
ZK3O402wf1c
|
a -x 2y and maybe
a -z as a -- oh,
|
ZK3O402wf1c
|
let me make that a minus one
and, just for variety let me
|
ZK3O402wf1c
|
take, -3z, -3ys, I should
keep the ys in that line,
|
ZK3O402wf1c
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and 4zs is, say, 4.
|
ZK3O402wf1c
|
Okay.
|
ZK3O402wf1c
|
That's three equations.
|
ZK3O402wf1c
|
I'm in three
dimensions, x, y, z.
|
ZK3O402wf1c
|
And, I don't have
a solution yet.
|
ZK3O402wf1c
|
So I want to understand the
equations and then solve them.
|
ZK3O402wf1c
|
Okay.
|
ZK3O402wf1c
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So how do I you understand them?
|
ZK3O402wf1c
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The row picture one way.
|
ZK3O402wf1c
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The column picture is
another very important way.
|
ZK3O402wf1c
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Just let's remember
the matrix form, here,
|
ZK3O402wf1c
|
because that's easy.
|
ZK3O402wf1c
|
The matrix form --
what's our matrix A?
|
ZK3O402wf1c
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Our matrix A is this right-hand
side, the two and the minus one
|
ZK3O402wf1c
|
and the zero from the first
row, the minus one and the two
|
ZK3O402wf1c
|
and the minus one
from the second row,
|
ZK3O402wf1c
|
the zero, the minus three and
the four from the third row.
|
ZK3O402wf1c
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So it's a three by three matrix.
|
ZK3O402wf1c
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Three equations, three unknowns.
|
ZK3O402wf1c
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And what's our right-hand side?
|
ZK3O402wf1c
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Of course, it's the vector,
zero minus one, four.
|
ZK3O402wf1c
|
Okay.
|
ZK3O402wf1c
|
So that's the way, well, that's
the short-hand to write out
|
ZK3O402wf1c
|
the three equations.
|
ZK3O402wf1c
|
But it's the picture that
I'm looking for today.
|
ZK3O402wf1c
|
Okay, so the row picture.
|
ZK3O402wf1c
|
All right, so I'm in
three dimensions, x,
|
ZK3O402wf1c
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find out when there
isn't a solution.
|
ZK3O402wf1c
|
y and z.
|
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