video_id stringclasses 5
values | text stringlengths 2 65 | start float64 0 4.67k | duration float64 0.12 26.1 |
|---|---|---|---|
ZK3O402wf1c | happen? | 735.72 | 0.56 |
ZK3O402wf1c | Okay, so how do I picture
this linear combination? | 736.28 | 5.56 |
ZK3O402wf1c | So I start with this vector
that's already here -- | 741.84 | 3.34 |
ZK3O402wf1c | so that's one of column
one, that's one times column | 745.18 | 3.04 |
ZK3O402wf1c | one, right there. | 748.22 | 2.18 |
ZK3O402wf1c | And now I want to add on --
so I'm going to hook the next | 750.4 | 3.32 |
ZK3O402wf1c | vector onto the front of the
arrow will start the next | 753.72 | 4.94 |
ZK3O402wf1c | vector and it will go this way. | 758.66 | 1.93 |
ZK3O402wf1c | So let's see, can I do it right? | 760.59 | 3.58 |
ZK3O402wf1c | If I added on one
of these vectors, | 764.17 | 2.07 |
ZK3O402wf1c | it would go left one and up two,
so we'd go left one and up two, | 766.24 | 4.349 |
ZK3O402wf1c | so it would probably
get us to there. | 770.589 | 1.541 |
ZK3O402wf1c | Maybe I'll do dotted
line for that. | 772.13 | 2.52 |
ZK3O402wf1c | Okay? | 774.65 | 0.96 |
ZK3O402wf1c | That's one of column
two tucked onto the end, | 775.61 | 4.1 |
ZK3O402wf1c | but I wanted to tuck
on two of column two. | 779.71 | 3.68 |
ZK3O402wf1c | So that -- the second one --
we'll go up left one and up two | 783.39 | 3.61 |
ZK3O402wf1c | also. | 787 | 0.66 |
ZK3O402wf1c | It'll probably end there. | 787.66 | 1.78 |
ZK3O402wf1c | And there's another one. | 789.44 | 1.08 |
ZK3O402wf1c | So what I've put in here
is two of column two. | 790.52 | 3.83 |
ZK3O402wf1c | Added on. | 797.31 | 2.03 |
ZK3O402wf1c | And where did I end up? | 799.34 | 1.275 |
ZK3O402wf1c | What are the coordinates
of this result? | 803.28 | 4.95 |
ZK3O402wf1c | What do I get when I take
one of this plus two of that? | 808.23 | 4.34 |
ZK3O402wf1c | I do get that, of course. | 812.57 | 2.24 |
ZK3O402wf1c | There it is, x is zero,
y is three, that's b. | 814.81 | 4.06 |
ZK3O402wf1c | That's the answer we wanted. | 818.87 | 2.66 |
ZK3O402wf1c | And how do I do it? | 821.53 | 2 |
ZK3O402wf1c | You see I do it just
like the first component. | 823.53 | 3.33 |
ZK3O402wf1c | I have a two and a minus
two that produces a zero, | 826.86 | 3.84 |
ZK3O402wf1c | and in the second component I
have a minus one and a four, | 830.7 | 3.98 |
ZK3O402wf1c | they combine to give the three. | 834.68 | 3.07 |
ZK3O402wf1c | But look at this picture. | 837.75 | 3.44 |
ZK3O402wf1c | So here's our key picture. | 841.19 | 2.04 |
ZK3O402wf1c | I combine this column and
this column to get this guy. | 843.23 | 7.05 |
ZK3O402wf1c | That was the b. | 850.28 | 1.08 |
ZK3O402wf1c | That's the zero three. | 851.36 | 2.32 |
ZK3O402wf1c | Okay. | 853.68 | 0.86 |
ZK3O402wf1c | So that idea of linear
combination is crucial, | 854.54 | 4.25 |
ZK3O402wf1c | and also -- | 858.79 | 2.34 |
ZK3O402wf1c | do we want to think
about this question? | 861.13 | 1.785 |
ZK3O402wf1c | Sure, why not. | 865.42 | 1.05 |
ZK3O402wf1c | What are all the combinations? | 866.47 | 3.36 |
ZK3O402wf1c | If I took -- can I
go back to xs and ys? | 869.83 | 2.825 |
ZK3O402wf1c | This is a question for really -- | 875.24 | 3.37 |
ZK3O402wf1c | it's going to come
up over and over, | 878.61 | 2.84 |
ZK3O402wf1c | but why don't we
see it once now? | 881.45 | 3.9 |
ZK3O402wf1c | If I took all the xs and all
the ys, all the combinations, | 885.35 | 5.26 |
ZK3O402wf1c | what would be all the results? | 890.61 | 3.22 |
ZK3O402wf1c | And, actually, the
result would be | 893.83 | 2.37 |
ZK3O402wf1c | that I could get any
right-hand side at all. | 896.2 | 3.2 |
ZK3O402wf1c | The combinations
of this and this | 899.4 | 3.28 |
ZK3O402wf1c | would fill the whole plane. | 902.68 | 3.57 |
ZK3O402wf1c | You can tuck that away. | 906.25 | 2.26 |
ZK3O402wf1c | We'll, explore it further. | 908.51 | 5.57 |
ZK3O402wf1c | But this idea of what linear
combination gives b and what do | 914.08 | 5.53 |
ZK3O402wf1c | all the linear
combinations give, | 919.61 | 2.04 |
ZK3O402wf1c | what are all the possible,
achievable right-hand sides be | 921.65 | 3.42 |
ZK3O402wf1c | -- that's going to be basic. | 925.07 | 1.54 |
ZK3O402wf1c | Okay. | 926.61 | 0.61 |
ZK3O402wf1c | Can I move to three
equations and three unknowns? | 927.22 | 4.76 |
ZK3O402wf1c | Because it's easy to
picture the two by two case. | 931.98 | 6.63 |
ZK3O402wf1c | Let me do a three
by three example. | 938.61 | 2.37 |
ZK3O402wf1c | Okay, I'll sort of
start it the same way, | 940.98 | 2.35 |
ZK3O402wf1c | say maybe 2x-y and maybe I'll
take no zs as a zero and maybe | 943.33 | 6.88 |
ZK3O402wf1c | a -x 2y and maybe
a -z as a -- oh, | 950.21 | 5.73 |
ZK3O402wf1c | let me make that a minus one
and, just for variety let me | 955.94 | 5.19 |
ZK3O402wf1c | take, -3z, -3ys, I should
keep the ys in that line, | 961.13 | 8.8 |
ZK3O402wf1c | and 4zs is, say, 4. | 969.93 | 4.47 |
ZK3O402wf1c | Okay. | 974.4 | 2.08 |
ZK3O402wf1c | That's three equations. | 976.48 | 2.62 |
ZK3O402wf1c | I'm in three
dimensions, x, y, z. | 979.1 | 3.28 |
ZK3O402wf1c | And, I don't have
a solution yet. | 982.38 | 4.83 |
ZK3O402wf1c | So I want to understand the
equations and then solve them. | 987.21 | 5.02 |
ZK3O402wf1c | Okay. | 992.23 | 0.55 |
ZK3O402wf1c | So how do I you understand them? | 992.78 | 2.67 |
ZK3O402wf1c | The row picture one way. | 995.45 | 1.9 |
ZK3O402wf1c | The column picture is
another very important way. | 997.35 | 3.77 |
ZK3O402wf1c | Just let's remember
the matrix form, here, | 1,001.12 | 2.6 |
ZK3O402wf1c | because that's easy. | 1,003.72 | 1.3 |
ZK3O402wf1c | The matrix form --
what's our matrix A? | 1,005.02 | 4.12 |
ZK3O402wf1c | Our matrix A is this right-hand
side, the two and the minus one | 1,009.14 | 5.14 |
ZK3O402wf1c | and the zero from the first
row, the minus one and the two | 1,014.28 | 4.34 |
ZK3O402wf1c | and the minus one
from the second row, | 1,018.62 | 2.16 |
ZK3O402wf1c | the zero, the minus three and
the four from the third row. | 1,020.78 | 7.75 |
ZK3O402wf1c | So it's a three by three matrix. | 1,028.53 | 2.52 |
ZK3O402wf1c | Three equations, three unknowns. | 1,031.05 | 1.92 |
ZK3O402wf1c | And what's our right-hand side? | 1,032.97 | 1.71 |
ZK3O402wf1c | Of course, it's the vector,
zero minus one, four. | 1,034.68 | 5.51 |
ZK3O402wf1c | Okay. | 1,040.19 | 0.99 |
ZK3O402wf1c | So that's the way, well, that's
the short-hand to write out | 1,041.18 | 5.77 |
ZK3O402wf1c | the three equations. | 1,046.95 | 1.74 |
ZK3O402wf1c | But it's the picture that
I'm looking for today. | 1,048.69 | 2.739 |
ZK3O402wf1c | Okay, so the row picture. | 1,051.429 | 1.041 |
ZK3O402wf1c | All right, so I'm in
three dimensions, x, | 1,052.47 | 5.9 |
ZK3O402wf1c | find out when there
isn't a solution. | 1,058.37 | 4.31 |
ZK3O402wf1c | y and z. | 1,062.68 | 3.27 |
ZK3O402wf1c | And I want to take those
equations one at a time and ask | 1,065.95 | 5.2 |
ZK3O402wf1c | -- | 1,071.15 | 0.67 |
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