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ZK3O402wf1c | It's because we already know
that that's the right x and y, |
ZK3O402wf1c | so why not take the correct
combination here and see it |
ZK3O402wf1c | happen? |
ZK3O402wf1c | Okay, so how do I picture
this linear combination? |
ZK3O402wf1c | So I start with this vector
that's already here -- |
ZK3O402wf1c | so that's one of column
one, that's one times column |
ZK3O402wf1c | 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 | so it would probably
get us to there. |
ZK3O402wf1c | Maybe I'll do dotted
line for that. |
ZK3O402wf1c | Okay? |
ZK3O402wf1c | That's one of column
two tucked onto the end, |
ZK3O402wf1c | but I wanted to tuck
on two of column two. |
ZK3O402wf1c | So that -- the second one --
we'll go up left one and up two |
ZK3O402wf1c | also. |
ZK3O402wf1c | It'll probably end there. |
ZK3O402wf1c | And there's another one. |
ZK3O402wf1c | So what I've put in here
is two of column two. |
ZK3O402wf1c | 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 | 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 | Sure, why not. |
ZK3O402wf1c | 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 | what would be all the results? |
ZK3O402wf1c | And, actually, the
result would be |
ZK3O402wf1c | that I could get any
right-hand side at all. |
ZK3O402wf1c | 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 | what are all the possible,
achievable right-hand sides be |
ZK3O402wf1c | -- 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 | 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 | 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 | So how do I you understand them? |
ZK3O402wf1c | The row picture one way. |
ZK3O402wf1c | The column picture is
another very important way. |
ZK3O402wf1c | Just let's remember
the matrix form, here, |
ZK3O402wf1c | because that's easy. |
ZK3O402wf1c | The matrix form --
what's our matrix A? |
ZK3O402wf1c | 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 | So it's a three by three matrix. |
ZK3O402wf1c | Three equations, three unknowns. |
ZK3O402wf1c | And what's our right-hand side? |
ZK3O402wf1c | 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 | find out when there
isn't a solution. |
ZK3O402wf1c | y and z. |
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