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ZK3O402wf1c | Do we go through
the origin or not? |
ZK3O402wf1c | In this case, yes, because
there's a zero over there. |
ZK3O402wf1c | In this case we don't
go through the origin, |
ZK3O402wf1c | because if x and y are
zero, we don't get three. |
ZK3O402wf1c | So, let me again say
suppose y is zero, |
ZK3O402wf1c | what x do we actually get? |
ZK3O402wf1c | If y is zero, then I
get x is minus three. |
ZK3O402wf1c | So if y is zero, I
go along minus three. |
ZK3O402wf1c | So there's one point
on this second line. |
ZK3O402wf1c | Now let me say, well,
suppose x is minus one -- |
ZK3O402wf1c | just to take another x. |
ZK3O402wf1c | If x is minus one,
then this is a one |
ZK3O402wf1c | and I think y should be a one,
because if x is minus one, |
ZK3O402wf1c | then I think y should be a
one and we'll get that point. |
ZK3O402wf1c | Is that right? |
ZK3O402wf1c | If x is minus one, that's a one. |
ZK3O402wf1c | If y is a one, that's
a two and the one |
ZK3O402wf1c | and the two make three and
that point's on the equation. |
ZK3O402wf1c | Okay. |
ZK3O402wf1c | Now, I should just
draw the line, right, |
ZK3O402wf1c | connecting those
two points at -- |
ZK3O402wf1c | that will give me
the whole line. |
ZK3O402wf1c | And if I've done
this reasonably well, |
ZK3O402wf1c | I think it's going to happen
to go through -- well, |
ZK3O402wf1c | not happen -- it was arranged
to go through that point. |
ZK3O402wf1c | So I think that the
second line is this one, |
ZK3O402wf1c | and this is the all-important
point that lies on both lines. |
ZK3O402wf1c | Shall we just check
that that point which |
ZK3O402wf1c | is the point x equal one
and y was two, right? |
ZK3O402wf1c | That's the point there
and that, I believe, |
ZK3O402wf1c | solves both equations. |
ZK3O402wf1c | Let's just check this. |
ZK3O402wf1c | If x is one, I have a minus one
plus four equals three, okay. |
ZK3O402wf1c | Apologies for
drawing this picture |
ZK3O402wf1c | that you've seen before. |
ZK3O402wf1c | But this -- seeing
the row picture -- |
ZK3O402wf1c | first of all, for n equal 2,
two equations and two unknowns, |
ZK3O402wf1c | it's the right place to start. |
ZK3O402wf1c | Okay. |
ZK3O402wf1c | So we've got the solution. |
ZK3O402wf1c | The point that
lies on both lines. |
ZK3O402wf1c | Now can I come to
the column picture? |
ZK3O402wf1c | Pay attention, this
is the key point. |
ZK3O402wf1c | So the column picture. |
ZK3O402wf1c | I'm now going to look at
the columns of the matrix. |
ZK3O402wf1c | I'm going to look at
this part and this part. |
ZK3O402wf1c | I'm going to say that the
x part is really x times -- |
ZK3O402wf1c | you see, I'm putting the two -- |
ZK3O402wf1c | I'm kind of getting the
two equations at once -- |
ZK3O402wf1c | that part and then I have a
y and in the first equation |
ZK3O402wf1c | it's multiplying a minus one and
in the second equation a two, |
ZK3O402wf1c | and on the right-hand
side, zero and three. |
ZK3O402wf1c | You see, the columns of the
matrix, the columns of A |
ZK3O402wf1c | are here and the
right-hand side b is there. |
ZK3O402wf1c | And now what is the
equation asking for? |
ZK3O402wf1c | It's asking us to find --
somehow to combine that vector |
ZK3O402wf1c | and this one in the right
amounts to get that one. |
ZK3O402wf1c | It's asking us to find the
right linear combination -- |
ZK3O402wf1c | this is called a
linear combination. |
ZK3O402wf1c | And it's the most fundamental
operation in the whole course. |
ZK3O402wf1c | It's a linear combination
of the columns. |
ZK3O402wf1c | That's what we're
seeing on the left side. |
ZK3O402wf1c | Again, I don't want to
write down a big definition. |
ZK3O402wf1c | You can see what it is. |
ZK3O402wf1c | There's column one,
there's column two. |
ZK3O402wf1c | I multiply by some
numbers and I add. |
ZK3O402wf1c | That's a combination -- a linear
combination and I want to make |
ZK3O402wf1c | those numbers the right
numbers to produce zero three. |
ZK3O402wf1c | Okay. |
ZK3O402wf1c | Now I want to draw a picture
that, represents what this -- |
ZK3O402wf1c | this is algebra. |
ZK3O402wf1c | What's the geometry, what's
the picture that goes with it? |
ZK3O402wf1c | Okay. |
ZK3O402wf1c | So again, these vectors
have two components, |
ZK3O402wf1c | so I better draw a
picture like that. |
ZK3O402wf1c | So can I put down these columns? |
ZK3O402wf1c | I'll draw these
columns as they are, |
ZK3O402wf1c | and then I'll do a
combination of them. |
ZK3O402wf1c | So the first column is over
two and down one, right? |
ZK3O402wf1c | So there's the first column. |
ZK3O402wf1c | The first column. |
ZK3O402wf1c | Column one. |
ZK3O402wf1c | It's the vector two minus one. |
ZK3O402wf1c | The second column is -- |
ZK3O402wf1c | minus one is the first
component and up two. |
ZK3O402wf1c | It's here. |
ZK3O402wf1c | There's column two. |
ZK3O402wf1c | So this, again, you see
what its components are. |
ZK3O402wf1c | Its components are
minus one, two. |
ZK3O402wf1c | Good. |
ZK3O402wf1c | That's this guy. |
ZK3O402wf1c | Now I have to take
a combination. |
ZK3O402wf1c | What combination shall I take? |
ZK3O402wf1c | Why not the right
combination, what the hell? |
ZK3O402wf1c | Okay. |
ZK3O402wf1c | So the combination
I'm going to take |
ZK3O402wf1c | is the right one to
produce zero three |
ZK3O402wf1c | and then we'll see it
happen in the picture. |
ZK3O402wf1c | So the right combination is
to take x as one of those |
ZK3O402wf1c | and two of these. |
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