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ZK3O402wf1c
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So this was the first component.
|
ZK3O402wf1c
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The second component
was a minus one.
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ZK3O402wf1c
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Maybe up here.
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ZK3O402wf1c
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That's column three, that's the
column zero minus one and four.
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ZK3O402wf1c
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This guy.
|
ZK3O402wf1c
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So, again, what's my problem?
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ZK3O402wf1c
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What this equation
is asking me to do
|
ZK3O402wf1c
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is to combine
these three vectors
|
ZK3O402wf1c
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with a right combination
to produce this one.
|
ZK3O402wf1c
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Well, you can see what the
right combination is, because
|
ZK3O402wf1c
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in this special problem,
specially chosen
|
ZK3O402wf1c
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by the lecturer, that right-hand
side that I'm trying to get
|
ZK3O402wf1c
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is actually one
of these columns.
|
ZK3O402wf1c
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So I know how to get that one.
|
ZK3O402wf1c
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So what's the solution?
|
ZK3O402wf1c
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What combination will work?
|
ZK3O402wf1c
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I just want one of
these and none of these.
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ZK3O402wf1c
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So x should be zero, y
should be zero and z should
|
ZK3O402wf1c
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be one.
|
ZK3O402wf1c
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That's the combination.
|
ZK3O402wf1c
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One of those is
obviously the right one.
|
ZK3O402wf1c
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Column three is
actually the same
|
ZK3O402wf1c
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as b in this particular problem.
|
ZK3O402wf1c
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I made it work
that way just so we
|
ZK3O402wf1c
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would get an answer,
(0,0,1), so somehow that's
|
ZK3O402wf1c
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the point where those
three planes met
|
ZK3O402wf1c
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and I couldn't see it before.
|
ZK3O402wf1c
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Of course, I won't always be
able to see it from the column
|
ZK3O402wf1c
|
picture, either.
|
ZK3O402wf1c
|
It's the next lecture, actually,
which is about elimination,
|
ZK3O402wf1c
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which is the systematic
way that everybody --
|
ZK3O402wf1c
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every bit of software, too --
|
ZK3O402wf1c
|
production, large-scale software
would solve the equations.
|
ZK3O402wf1c
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So the lecture that's coming up.
|
ZK3O402wf1c
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If I was to add that
to the syllabus,
|
ZK3O402wf1c
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will be about how to find
x, y, z in all cases.
|
ZK3O402wf1c
|
Can I just think again,
though, about the big picture?
|
ZK3O402wf1c
|
By the big picture I mean
let's keep this same matrix
|
ZK3O402wf1c
|
on the left but
imagine that we have
|
ZK3O402wf1c
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a different right-hand side.
|
ZK3O402wf1c
|
Oh, let me take a
different right-hand side.
|
ZK3O402wf1c
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So I'll change that
right-hand side
|
ZK3O402wf1c
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to something that actually
is also pretty special.
|
ZK3O402wf1c
|
Let me change it to --
|
ZK3O402wf1c
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if I add those
first two columns,
|
ZK3O402wf1c
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that would give me a one
and a one and a minus three.
|
ZK3O402wf1c
|
There's a very special
right-hand side.
|
ZK3O402wf1c
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I just cooked it up by
adding this one to this one.
|
ZK3O402wf1c
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Now, what's the solution with
this new right-hand side?
|
ZK3O402wf1c
|
The solution with this new
right-hand side is clear.
|
ZK3O402wf1c
|
took one of these
and none of those.
|
ZK3O402wf1c
|
So actually, it just
changed around to this
|
ZK3O402wf1c
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when I took this
new right-hand side.
|
ZK3O402wf1c
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Okay.
|
ZK3O402wf1c
|
So in the row picture, I
have three different planes,
|
ZK3O402wf1c
|
three new planes meeting
now at this point.
|
ZK3O402wf1c
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In the column picture, I
have the same three columns,
|
ZK3O402wf1c
|
but now I'm combining
them to produce this guy,
|
ZK3O402wf1c
|
and it turned out that column
one plus column two which would
|
ZK3O402wf1c
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be somewhere -- there
is the right column --
|
ZK3O402wf1c
|
one of this and one of this
would give me the new b.
|
ZK3O402wf1c
|
Okay.
|
ZK3O402wf1c
|
So we squeezed in
an extra example.
|
ZK3O402wf1c
|
But now think about all
bs, all right-hand sides.
|
ZK3O402wf1c
|
Can I solve these equations
for every right-hand side?
|
ZK3O402wf1c
|
Can I ask that question?
|
ZK3O402wf1c
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So that's the algebra question.
|
ZK3O402wf1c
|
Can I solve A x=b for every b?
|
ZK3O402wf1c
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Let me write that down.
|
ZK3O402wf1c
|
Can I solve A x =b for
every right-hand side b?
|
ZK3O402wf1c
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I mean, is there a solution?
|
ZK3O402wf1c
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And then, if there
is, elimination
|
ZK3O402wf1c
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will give me a way to find it.
|
ZK3O402wf1c
|
I really wanted to ask,
is there a solution
|
ZK3O402wf1c
|
for every right-hand side?
|
ZK3O402wf1c
|
So now, can I put that
in different words --
|
ZK3O402wf1c
|
in this linear
combination words?
|
ZK3O402wf1c
|
So in linear combination words,
do the linear combinations
|
ZK3O402wf1c
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of the columns fill
three dimensional space?
|
ZK3O402wf1c
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Every b means all the bs
in three dimensional space.
|
ZK3O402wf1c
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Do you see that I'm just
asking the same question
|
ZK3O402wf1c
|
in different words?
|
ZK3O402wf1c
|
Solving A x --
|
ZK3O402wf1c
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A x -- that's very important.
|
ZK3O402wf1c
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A times x -- when I multiply
a matrix by a vector,
|
ZK3O402wf1c
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I get a combination
of the columns.
|
ZK3O402wf1c
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I'll write that
down in a moment.
|
ZK3O402wf1c
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But in my column picture,
that's really what I'm doing.
|
ZK3O402wf1c
|
I'm taking linear combinations
of these three columns
|
ZK3O402wf1c
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and I'm trying to find b.
|
ZK3O402wf1c
|
And, actually, the answer
for this matrix will be yes.
|
ZK3O402wf1c
|
For this matrix A -- for these
columns, the answer is yes.
|
ZK3O402wf1c
|
This matrix -- that I chose for
an example is a good matrix.
|
ZK3O402wf1c
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A non-singular matrix.
|
ZK3O402wf1c
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An invertible matrix.
|
ZK3O402wf1c
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Those will be the matrices
that we like best.
|
ZK3O402wf1c
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There could be other --
|
ZK3O402wf1c
|
and we will see other matrices
where the answer becomes, no --
|
ZK3O402wf1c
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oh, actually, you can see
when it would become no.
|
ZK3O402wf1c
|
What could go wrong? find out
-- because if elimination fails,
|
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