video_id stringclasses 5
values | text stringlengths 2 65 | start float64 0 4.67k | duration float64 0.12 26.1 |
|---|---|---|---|
ZK3O402wf1c | In this case we don't
go through the origin, | 390.02 | 2.25 |
ZK3O402wf1c | because if x and y are
zero, we don't get three. | 392.27 | 3.95 |
ZK3O402wf1c | So, let me again say
suppose y is zero, | 396.22 | 4.13 |
ZK3O402wf1c | what x do we actually get? | 400.35 | 2.21 |
ZK3O402wf1c | If y is zero, then I
get x is minus three. | 402.56 | 5.67 |
ZK3O402wf1c | So if y is zero, I
go along minus three. | 408.23 | 4.38 |
ZK3O402wf1c | So there's one point
on this second line. | 412.61 | 3.75 |
ZK3O402wf1c | Now let me say, well,
suppose x is minus one -- | 416.36 | 7.09 |
ZK3O402wf1c | just to take another x. | 423.45 | 1.92 |
ZK3O402wf1c | If x is minus one,
then this is a one | 425.37 | 3.91 |
ZK3O402wf1c | and I think y should be a one,
because if x is minus one, | 429.28 | 6.59 |
ZK3O402wf1c | then I think y should be a
one and we'll get that point. | 435.87 | 3.92 |
ZK3O402wf1c | Is that right? | 439.79 | 1.11 |
ZK3O402wf1c | If x is minus one, that's a one. | 440.9 | 2.43 |
ZK3O402wf1c | If y is a one, that's
a two and the one | 443.33 | 2.31 |
ZK3O402wf1c | and the two make three and
that point's on the equation. | 445.64 | 2.91 |
ZK3O402wf1c | Okay. | 448.55 | 0.78 |
ZK3O402wf1c | Now, I should just
draw the line, right, | 449.33 | 2.41 |
ZK3O402wf1c | connecting those
two points at -- | 451.74 | 2.73 |
ZK3O402wf1c | that will give me
the whole line. | 454.47 | 1.94 |
ZK3O402wf1c | And if I've done
this reasonably well, | 456.41 | 4.07 |
ZK3O402wf1c | I think it's going to happen
to go through -- well, | 460.48 | 2.16 |
ZK3O402wf1c | not happen -- it was arranged
to go through that point. | 462.64 | 3.88 |
ZK3O402wf1c | So I think that the
second line is this one, | 466.52 | 4.08 |
ZK3O402wf1c | and this is the all-important
point that lies on both lines. | 470.6 | 5.5 |
ZK3O402wf1c | Shall we just check
that that point which | 476.1 | 2.73 |
ZK3O402wf1c | is the point x equal one
and y was two, right? | 478.83 | 4.86 |
ZK3O402wf1c | That's the point there
and that, I believe, | 483.69 | 4.1 |
ZK3O402wf1c | solves both equations. | 487.79 | 2.74 |
ZK3O402wf1c | Let's just check this. | 490.53 | 0.97 |
ZK3O402wf1c | If x is one, I have a minus one
plus four equals three, okay. | 491.5 | 7.09 |
ZK3O402wf1c | Apologies for
drawing this picture | 498.59 | 3.09 |
ZK3O402wf1c | that you've seen before. | 501.68 | 3.47 |
ZK3O402wf1c | But this -- seeing
the row picture -- | 505.15 | 4.54 |
ZK3O402wf1c | first of all, for n equal 2,
two equations and two unknowns, | 509.69 | 3.45 |
ZK3O402wf1c | it's the right place to start. | 513.14 | 2.32 |
ZK3O402wf1c | Okay. | 515.46 | 0.81 |
ZK3O402wf1c | So we've got the solution. | 516.27 | 1.42 |
ZK3O402wf1c | The point that
lies on both lines. | 517.69 | 1.989 |
ZK3O402wf1c | Now can I come to
the column picture? | 519.679 | 3.921 |
ZK3O402wf1c | Pay attention, this
is the key point. | 523.6 | 4.28 |
ZK3O402wf1c | So the column picture. | 527.88 | 2.05 |
ZK3O402wf1c | I'm now going to look at
the columns of the matrix. | 529.93 | 3.59 |
ZK3O402wf1c | I'm going to look at
this part and this part. | 533.52 | 3.27 |
ZK3O402wf1c | I'm going to say that the
x part is really x times -- | 536.79 | 8.22 |
ZK3O402wf1c | you see, I'm putting the two -- | 545.01 | 1.73 |
ZK3O402wf1c | I'm kind of getting the
two equations at once -- | 546.74 | 3.42 |
ZK3O402wf1c | that part and then I have a
y and in the first equation | 550.16 | 5.13 |
ZK3O402wf1c | it's multiplying a minus one and
in the second equation a two, | 555.29 | 3.97 |
ZK3O402wf1c | and on the right-hand
side, zero and three. | 559.26 | 2.77 |
ZK3O402wf1c | You see, the columns of the
matrix, the columns of A | 566.5 | 3.36 |
ZK3O402wf1c | are here and the
right-hand side b is there. | 569.86 | 3.19 |
ZK3O402wf1c | And now what is the
equation asking for? | 573.05 | 3.89 |
ZK3O402wf1c | It's asking us to find --
somehow to combine that vector | 576.94 | 5.81 |
ZK3O402wf1c | and this one in the right
amounts to get that one. | 582.75 | 4.72 |
ZK3O402wf1c | It's asking us to find the
right linear combination -- | 587.47 | 4.36 |
ZK3O402wf1c | this is called a
linear combination. | 591.83 | 3.59 |
ZK3O402wf1c | And it's the most fundamental
operation in the whole course. | 595.42 | 3.42 |
ZK3O402wf1c | It's a linear combination
of the columns. | 601.71 | 4.62 |
ZK3O402wf1c | That's what we're
seeing on the left side. | 606.33 | 3.22 |
ZK3O402wf1c | Again, I don't want to
write down a big definition. | 609.55 | 3.64 |
ZK3O402wf1c | You can see what it is. | 613.19 | 1.82 |
ZK3O402wf1c | There's column one,
there's column two. | 615.01 | 2.46 |
ZK3O402wf1c | I multiply by some
numbers and I add. | 617.47 | 4.22 |
ZK3O402wf1c | That's a combination -- a linear
combination and I want to make | 621.69 | 4.25 |
ZK3O402wf1c | those numbers the right
numbers to produce zero three. | 625.94 | 4.93 |
ZK3O402wf1c | Okay. | 630.87 | 1.2 |
ZK3O402wf1c | Now I want to draw a picture
that, represents what this -- | 632.07 | 5.04 |
ZK3O402wf1c | this is algebra. | 637.11 | 1.12 |
ZK3O402wf1c | What's the geometry, what's
the picture that goes with it? | 638.23 | 3.51 |
ZK3O402wf1c | Okay. | 641.74 | 0.62 |
ZK3O402wf1c | So again, these vectors
have two components, | 642.36 | 2.9 |
ZK3O402wf1c | so I better draw a
picture like that. | 645.26 | 5.47 |
ZK3O402wf1c | So can I put down these columns? | 650.73 | 1.95 |
ZK3O402wf1c | I'll draw these
columns as they are, | 652.68 | 2.79 |
ZK3O402wf1c | and then I'll do a
combination of them. | 655.47 | 3.23 |
ZK3O402wf1c | So the first column is over
two and down one, right? | 658.7 | 5.3 |
ZK3O402wf1c | So there's the first column. | 664 | 2.21 |
ZK3O402wf1c | The first column. | 666.21 | 1.65 |
ZK3O402wf1c | Column one. | 667.86 | 3.1 |
ZK3O402wf1c | It's the vector two minus one. | 670.96 | 2.87 |
ZK3O402wf1c | The second column is -- | 673.83 | 4.75 |
ZK3O402wf1c | minus one is the first
component and up two. | 678.58 | 3.42 |
ZK3O402wf1c | It's here. | 682 | 2.96 |
ZK3O402wf1c | There's column two. | 684.96 | 2.75 |
ZK3O402wf1c | So this, again, you see
what its components are. | 687.71 | 4.57 |
ZK3O402wf1c | Its components are
minus one, two. | 692.28 | 3.25 |
ZK3O402wf1c | Good. | 695.53 | 0.5 |
ZK3O402wf1c | That's this guy. | 696.03 | 0.81 |
ZK3O402wf1c | Now I have to take
a combination. | 699.471 | 1.374 |
ZK3O402wf1c | What combination shall I take? | 703.74 | 3.57 |
ZK3O402wf1c | Why not the right
combination, what the hell? | 707.31 | 3.2 |
ZK3O402wf1c | Okay. | 710.51 | 0.5 |
ZK3O402wf1c | So the combination
I'm going to take | 711.01 | 2.83 |
ZK3O402wf1c | is the right one to
produce zero three | 713.84 | 3.41 |
ZK3O402wf1c | and then we'll see it
happen in the picture. | 717.25 | 2.37 |
ZK3O402wf1c | So the right combination is
to take x as one of those | 719.62 | 4.91 |
ZK3O402wf1c | and two of these. | 724.53 | 1.61 |
ZK3O402wf1c | It's because we already know
that that's the right x and y, | 729.23 | 3.53 |
ZK3O402wf1c | so why not take the correct
combination here and see it | 732.76 | 2.96 |
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