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