WEBVTT
00:00:09.840 --> 00:00:14.610
OK, this is linear
algebra lecture nine.
00:00:14.610 --> 00:00:21.130
And this is a key lecture, this
is where we get these ideas
00:00:21.130 --> 00:00:27.480
of linear independence,
when a bunch of vectors are
00:00:27.480 --> 00:00:29.600
independent --
00:00:29.600 --> 00:00:33.890
or dependent,
that's the opposite.
00:00:33.890 --> 00:00:36.680
The space they span.
00:00:36.680 --> 00:00:40.550
A basis for a subspace
or a basis for a vector
00:00:40.550 --> 00:00:42.730
space, that's a central idea.
00:00:42.730 --> 00:00:47.060
And then the dimension
of that subspace.
00:00:47.060 --> 00:00:50.240
So this is the day
that those words
00:00:50.240 --> 00:00:52.960
get assigned clear meanings.
00:00:52.960 --> 00:00:58.380
And emphasize that we talk
about a bunch of vectors
00:00:58.380 --> 00:00:59.550
being independent.
00:00:59.550 --> 00:01:02.780
Wouldn't talk about a
matrix being independent.
00:01:02.780 --> 00:01:05.150
A bunch of vectors
being independent.
00:01:05.150 --> 00:01:07.840
A bunch of vectors
spanning a space.
00:01:07.840 --> 00:01:11.480
A bunch of vectors
being a basis.
00:01:11.480 --> 00:01:15.070
And the dimension
is some number.
00:01:15.070 --> 00:01:17.480
OK, so what are the definitions?
00:01:17.480 --> 00:01:21.840
Can I begin with a fact,
a highly important fact,
00:01:21.840 --> 00:01:28.240
that, I didn't call directly
attention to earlier.
00:01:28.240 --> 00:01:35.900
Suppose I have a matrix and
I look at Ax equals zero.
00:01:35.900 --> 00:01:39.530
Suppose the matrix
has a lot of columns,
00:01:39.530 --> 00:01:43.340
so that n is bigger than m.
00:01:43.340 --> 00:01:46.320
So I'm looking at n equations --
00:01:46.320 --> 00:01:48.920
I mean, sorry, m
equations, a small number
00:01:48.920 --> 00:01:52.650
of equations m,
and more unknowns.
00:01:52.650 --> 00:01:55.960
I have more unknowns
than equations.
00:01:55.960 --> 00:01:57.040
Let me write that down.
00:01:57.040 --> 00:01:59.850
More unknowns than equations.
00:01:59.850 --> 00:02:03.170
More unknown x-s than equations.
00:02:03.170 --> 00:02:15.440
Then the conclusion
is that there's
00:02:15.440 --> 00:02:19.330
something in the null
space of A, other
00:02:19.330 --> 00:02:21.350
than just the zero vector.
00:02:21.350 --> 00:02:26.320
The conclusion is there
are some non-zero x-s
00:02:26.320 --> 00:02:28.220
such that Ax is zero.
00:02:28.220 --> 00:02:30.030
There are some
special solutions.
00:02:30.030 --> 00:02:31.060
And why?
00:02:31.060 --> 00:02:33.590
We know why.
00:02:33.590 --> 00:02:38.120
I mean, it sort of like seems
like a reasonable thing, more
00:02:38.120 --> 00:02:42.150
unknowns than equations,
then it seems reasonable
00:02:42.150 --> 00:02:43.540
that we can solve them.
00:02:43.540 --> 00:02:49.210
But we have a, a clear algorithm
which starts with a system
00:02:49.210 --> 00:02:55.440
and does elimination, gets
the thing into an echelon
00:02:55.440 --> 00:03:01.630
form with some pivots
and pivot columns,
00:03:01.630 --> 00:03:06.050
and possibly some free columns
that don't have pivots.
00:03:06.050 --> 00:03:09.290
And the point is here there
will be some free columns.
00:03:09.290 --> 00:03:14.940
The reason, so the
reason is there must --
00:03:14.940 --> 00:03:20.080
there will be free
variables, at least one.
00:03:25.010 --> 00:03:27.580
That's the reason.
00:03:27.580 --> 00:03:30.130
That we now have this --
00:03:30.130 --> 00:03:36.200
a complete, algorithm, a
complete systematic way to say,
00:03:36.200 --> 00:03:38.860
OK, we take the
system Ax equals zero,
00:03:38.860 --> 00:03:42.370
we row reduce, we identify
the free variables,
00:03:42.370 --> 00:03:48.690
and, since there are n
variables and at most m pivots,
00:03:48.690 --> 00:03:51.840
there will be some free
variables, at least one,
00:03:51.840 --> 00:03:55.090
at least n-m in fact, left over.
00:03:55.090 --> 00:04:00.480
And those variables I can
assign non-zero values to.
00:04:00.480 --> 00:04:02.110
I don't have to
set those to zero.
00:04:02.110 --> 00:04:04.580
I can take them to be
one or whatever I like,
00:04:04.580 --> 00:04:07.970
and then I can solve
for the pivot variables.
00:04:07.970 --> 00:04:11.870
So then it gives me a
solution to Ax equals zero.
00:04:11.870 --> 00:04:15.820
And it's a solution
that isn't all zeros.
00:04:15.820 --> 00:04:21.769
So, that's an important
point that we'll
00:04:21.769 --> 00:04:26.350
use now in this lecture.
00:04:26.350 --> 00:04:30.110
So now I want to say what does
it mean for a bunch of vectors
00:04:30.110 --> 00:04:31.470
to be independent.
00:04:31.470 --> 00:04:32.370
OK.
00:04:32.370 --> 00:04:36.276
So this is like the
background that we know.
00:04:36.276 --> 00:04:37.900
Now I want to speak
about independence.
00:04:41.500 --> 00:04:42.000
OK.
00:04:46.110 --> 00:04:48.740
Let's see.
00:04:48.740 --> 00:04:55.430
I can give you the abstract
definition, and I will,
00:04:55.430 --> 00:05:00.325
but I would also like to
give you the direct meaning.
00:05:03.700 --> 00:05:12.700
So the question is, when
vectors x1, x2 up to --
00:05:12.700 --> 00:05:19.860
Suppose I have n vectors
are independent if.
00:05:19.860 --> 00:05:22.350
Now I have to give you --
00:05:22.350 --> 00:05:25.560
or linearly independent --
00:05:25.560 --> 00:05:30.470
I'll often just say and
write independent for short.
00:05:30.470 --> 00:05:31.300
OK.
00:05:31.300 --> 00:05:33.580
I'll give you the
full definition.
00:05:33.580 --> 00:05:36.920
These are just vectors
in some vector space.
00:05:36.920 --> 00:05:39.360
I can take combinations of them.
00:05:39.360 --> 00:05:45.070
The question is, do any
combinations give zero?
00:05:45.070 --> 00:05:47.210
If some combination
of those vectors
00:05:47.210 --> 00:05:51.030
gives the zero vector,
other than the combination
00:05:51.030 --> 00:05:55.520
of all zeros, then
they're dependent.
00:05:55.520 --> 00:06:06.100
They're independent if no
combination gives the zero
00:06:06.100 --> 00:06:08.550
vector --
00:06:08.550 --> 00:06:17.070
and then I have, I'll have
to put in an except the zero
00:06:17.070 --> 00:06:18.840
combination.
00:06:18.840 --> 00:06:21.160
So what do I mean by that?
00:06:21.160 --> 00:06:23.720
No combination gives
the zero vector.
00:06:23.720 --> 00:06:29.950
Any combination
c1 x1+c2 x2 plus,
00:06:29.950 --> 00:06:39.260
plus cn xn is not zero except
for the zero combination.
00:06:39.260 --> 00:06:45.610
This is when all the c-s,
all the c-s are zero.
00:06:45.610 --> 00:06:46.315
Then of course.
00:06:49.290 --> 00:06:52.100
That combination --
I know I'll get zero.
00:06:52.100 --> 00:06:55.950
But the question is, does any
other combination give zero?
00:06:55.950 --> 00:06:59.970
If not, then the
vectors are independent.
00:06:59.970 --> 00:07:02.630
If some other combination
does give zero,
00:07:02.630 --> 00:07:04.880
the vectors are dependent.
00:07:04.880 --> 00:07:05.510
OK.
00:07:05.510 --> 00:07:08.420
Let's just take examples.
00:07:08.420 --> 00:07:14.510
Suppose I'm in, say, in
two dimensional space.
00:07:14.510 --> 00:07:15.590
OK.
00:07:15.590 --> 00:07:16.940
I give you --
00:07:16.940 --> 00:07:19.900
I'd like to first
take an example --
00:07:19.900 --> 00:07:24.480
let me take an example where
I have a vector and twice that
00:07:24.480 --> 00:07:26.290
vector.
00:07:26.290 --> 00:07:29.720
So that's two vectors, V and 2V.
00:07:29.720 --> 00:07:33.830
Are those dependent
or independent?
00:07:33.830 --> 00:07:35.820
Those are dependent
for sure, right,
00:07:35.820 --> 00:07:40.380
because there's one
vector is twice the other.
00:07:40.380 --> 00:07:42.250
One vector is twice
as long as the other,
00:07:42.250 --> 00:07:44.780
so if the word dependent
means anything,
00:07:44.780 --> 00:07:46.670
these should be dependent.
00:07:46.670 --> 00:07:47.910
And they are.
00:07:47.910 --> 00:07:51.860
And in fact, I would
take two of the first --
00:07:51.860 --> 00:07:56.750
so here's, here is a vector V
and the other guy is a vector
00:07:56.750 --> 00:07:59.350
2V, that's my --
00:07:59.350 --> 00:08:04.730
so there's a vector V1 and
my next vector V2 is 2V1.
00:08:04.730 --> 00:08:08.630
Of course those are
dependent, because two
00:08:08.630 --> 00:08:14.120
of these first vectors minus
the second vector is zero.
00:08:14.120 --> 00:08:17.620
That's a combination of these
two vectors that gives the zero
00:08:17.620 --> 00:08:18.120
vector.
00:08:18.120 --> 00:08:20.060
OK, that was clear.
00:08:20.060 --> 00:08:24.430
Suppose, suppose
I have a vector --
00:08:24.430 --> 00:08:25.060
here's another
00:08:25.060 --> 00:08:25.800
example.
00:08:25.800 --> 00:08:26.950
It's easy example.
00:08:26.950 --> 00:08:31.010
Suppose I have a vector and the
other guy is the zero vector.
00:08:31.010 --> 00:08:36.820
Suppose I have a vector V1
and V2 is the zero vector.
00:08:36.820 --> 00:08:43.309
Then are those vectors
dependent or independent?
00:08:43.309 --> 00:08:46.230
They're dependent again.
00:08:46.230 --> 00:08:50.480
You could say, well, this
guy is zero times that one.
00:08:50.480 --> 00:08:52.970
This one is some
combination of those.
00:08:52.970 --> 00:08:54.900
But let me write
it the other way.
00:08:54.900 --> 00:09:01.110
Let me say -- what combination,
how many V1s and how many V2s
00:09:01.110 --> 00:09:02.671
shall I take to get the zero
00:09:02.671 --> 00:09:03.170
vector?
00:09:05.810 --> 00:09:11.490
If, if V1 is like the vector two
one and V2 is the zero vector,
00:09:11.490 --> 00:09:15.740
zero zero, then I
would like to show
00:09:15.740 --> 00:09:18.640
that some combination of
those gives the zero vector.
00:09:18.640 --> 00:09:19.820
What shall I take?
00:09:19.820 --> 00:09:22.660
How many V1s shall I take?
00:09:22.660 --> 00:09:23.540
Zero of them.
00:09:23.540 --> 00:09:25.310
Yeah, no, take no V1s.
00:09:25.310 --> 00:09:28.290
But how many V2s?
00:09:28.290 --> 00:09:28.970
Six.
00:09:28.970 --> 00:09:31.700
OK.
00:09:31.700 --> 00:09:34.790
Or five.
00:09:34.790 --> 00:09:38.590
Then -- in other words,
the point is if the zero
00:09:38.590 --> 00:09:41.340
vector's in there,
if the zero --
00:09:41.340 --> 00:09:44.760
if one of these vectors
is the zero vector,
00:09:44.760 --> 00:09:47.490
independence is dead, right?
00:09:47.490 --> 00:09:50.440
If one of those vectors is the
zero vector then I could always
00:09:50.440 --> 00:09:51.970
take --
00:09:51.970 --> 00:09:54.120
include that one and
none of the others,
00:09:54.120 --> 00:09:59.370
and I would get the zero answer,
and I would show dependence.
00:09:59.370 --> 00:10:00.230
OK.
00:10:00.230 --> 00:10:04.890
Now, let me, let me
finally draw an example
00:10:04.890 --> 00:10:06.860
where they will be independent.
00:10:06.860 --> 00:10:10.900
Suppose that's V1 and that's V2.
00:10:10.900 --> 00:10:14.820
Those are surely
independent, right?
00:10:14.820 --> 00:10:19.400
Any combination of
V1 and V2, will not
00:10:19.400 --> 00:10:22.780
be zero except, the
zero combination.
00:10:22.780 --> 00:10:24.390
So those would be independent.
00:10:24.390 --> 00:10:28.040
But now let me, let me
stick in a third vector, V3.
00:10:31.100 --> 00:10:34.200
Independent or dependent
now, those three vectors?
00:10:34.200 --> 00:10:37.390
So now n is three here.
00:10:37.390 --> 00:10:42.150
I'm in two dimensional space,
whatever, I'm in the plane.
00:10:42.150 --> 00:10:47.390
I have three vectors that
I didn't draw so carefully.
00:10:47.390 --> 00:10:50.520
I didn't even tell you
what exactly they were.
00:10:50.520 --> 00:10:55.820
But what's this answer on
dependent or independent?
00:10:55.820 --> 00:10:57.200
Dependent.
00:10:57.200 --> 00:11:00.570
How do I know those
are dependent?
00:11:00.570 --> 00:11:05.570
How do I know that some
combination of V1, V2, and V3
00:11:05.570 --> 00:11:09.010
gives me the zero vector?
00:11:09.010 --> 00:11:12.130
I know because of that.
00:11:12.130 --> 00:11:20.160
That's the key
fact that tells me
00:11:20.160 --> 00:11:23.990
that three vectors in the
plane have to be dependent.
00:11:23.990 --> 00:11:25.120
Why's that?
00:11:25.120 --> 00:11:28.650
What's the connection between
the dependence of these three
00:11:28.650 --> 00:11:30.620
vectors and that fact?
00:11:30.620 --> 00:11:31.140
OK.
00:11:31.140 --> 00:11:33.350
So here's the connection.
00:11:33.350 --> 00:11:42.260
I take the matrix A that has
V1 in its first column, V2
00:11:42.260 --> 00:11:46.560
in its second column,
V3 in its third column.
00:11:46.560 --> 00:11:48.560
So it's got three columns.
00:11:48.560 --> 00:11:49.600
And V1 --
00:11:49.600 --> 00:11:51.950
I don't know, that
looks like about two one
00:11:51.950 --> 00:11:52.880
to me.
00:11:52.880 --> 00:11:56.430
V2 looks like it
might be one two.
00:11:56.430 --> 00:12:01.990
V3 looks like it might be maybe
two, maybe two and a half,
00:12:01.990 --> 00:12:03.570
minus one.
00:12:07.520 --> 00:12:08.230
OK.
00:12:08.230 --> 00:12:13.900
Those are my three vectors, and
I put them in the columns of A.
00:12:13.900 --> 00:12:18.530
Now that matrix A
is two by three.
00:12:18.530 --> 00:12:22.960
It fits this pattern, that
where we know we've got extra
00:12:22.960 --> 00:12:25.900
variables, we know we
have some free variables,
00:12:25.900 --> 00:12:28.640
we know that there's
some combination --
00:12:28.640 --> 00:12:34.650
and let me instead of x-s, let
me call them c1, c2, and c3 --
00:12:34.650 --> 00:12:39.380
that gives the zero vector.
00:12:39.380 --> 00:12:41.830
Sorry that my little bit
of art got in the way.
00:12:41.830 --> 00:12:44.470
Do you see the point?
00:12:44.470 --> 00:12:48.600
When I have a matrix,
I'm interested
00:12:48.600 --> 00:12:53.710
in whether its columns are
dependent or independent.
00:12:53.710 --> 00:12:56.470
The columns are
dependent if there
00:12:56.470 --> 00:12:58.480
is something in the null space.
00:12:58.480 --> 00:13:00.860
The columns are
dependent because this,
00:13:00.860 --> 00:13:03.090
this thing in the
null space says
00:13:03.090 --> 00:13:09.190
that c1 of that plus c2 of
that plus c3 of this is zero.
00:13:09.190 --> 00:13:13.670
So in other words, I can go
out some V1, out some more V2,
00:13:13.670 --> 00:13:16.260
back on V3, and end up zero.
00:13:19.010 --> 00:13:19.850
OK.
00:13:19.850 --> 00:13:25.290
So let -- here I've give the
general, abstract definition,
00:13:25.290 --> 00:13:28.360
but let me repeat
that definition --
00:13:28.360 --> 00:13:31.840
this is like repeat --
00:13:31.840 --> 00:13:39.270
let me call them Vs now.
00:13:39.270 --> 00:13:47.450
V1 up to Vn are the
columns of a matrix A.
00:13:47.450 --> 00:13:50.120
In other words,
this is telling me
00:13:50.120 --> 00:13:57.510
that if I'm in m
dimensional space,
00:13:57.510 --> 00:14:00.630
like two dimensional
space in the example,
00:14:00.630 --> 00:14:03.940
I can answer the
dependence-independence
00:14:03.940 --> 00:14:10.180
question directly by
putting those vectors
00:14:10.180 --> 00:14:11.580
in the columns of a matrix.
00:14:14.330 --> 00:14:30.380
They are independent if the
null space of A, of A, is what?
00:14:33.230 --> 00:14:36.370
If I have a bunch of
columns in a matrix,
00:14:36.370 --> 00:14:38.870
I'm looking at
their combinations,
00:14:38.870 --> 00:14:44.850
but that's just A times
the vector of c-s.
00:14:44.850 --> 00:14:48.020
And these columns
will be independent
00:14:48.020 --> 00:14:56.390
if the null space of
A is the zero vector.
00:15:01.270 --> 00:15:11.960
They are dependent if there's
something else in there.
00:15:11.960 --> 00:15:17.380
If there's something else in
the null space, if A times c
00:15:17.380 --> 00:15:25.240
gives the zero vector
for some non-zero vector
00:15:25.240 --> 00:15:26.575
c in the null space.
00:15:29.160 --> 00:15:30.770
Then they're dependent,
because that's
00:15:30.770 --> 00:15:34.380
telling me a combination of the
columns gives the zero column.
00:15:34.380 --> 00:15:38.040
I think you're with
be, because we've seen,
00:15:38.040 --> 00:15:40.340
like, lecture after
lecture, we're
00:15:40.340 --> 00:15:43.500
looking at the combinations
of the columns and asking,
00:15:43.500 --> 00:15:45.550
do we get zero or don't we?
00:15:45.550 --> 00:15:49.030
And now we're giving
the official name,
00:15:49.030 --> 00:15:54.010
dependent if we do,
independent if we don't.
00:15:54.010 --> 00:15:57.890
So I could express this
in other words now.
00:15:57.890 --> 00:16:01.420
I could say the rank -- what's
the rank in this independent
00:16:01.420 --> 00:16:02.450
case?
00:16:02.450 --> 00:16:05.020
The rank r of the,
of the matrix,
00:16:05.020 --> 00:16:08.235
in the case of
independent columns, is?
00:16:11.570 --> 00:16:14.740
So the columns are independent.
00:16:14.740 --> 00:16:18.560
So how many pivot
columns have I got.
00:16:18.560 --> 00:16:19.850
All n.
00:16:19.850 --> 00:16:23.310
All the columns would
be pivot columns,
00:16:23.310 --> 00:16:25.980
because free columns
are telling me
00:16:25.980 --> 00:16:29.820
that they're a combination
of earlier columns.
00:16:29.820 --> 00:16:33.080
So this would be the
case where the rank is n.
00:16:33.080 --> 00:16:36.910
This would be the case where
the rank is smaller than n.
00:16:39.650 --> 00:16:44.770
So in this case the rank is
n and the null space of A
00:16:44.770 --> 00:16:48.750
is only the zero vector.
00:16:48.750 --> 00:16:50.800
And no free variables.
00:16:50.800 --> 00:16:52.400
No free variables.
00:16:56.520 --> 00:16:58.975
And this is the case
yes free variables.
00:17:04.300 --> 00:17:09.589
If you'll allow me to stretch
the English language that far.
00:17:09.589 --> 00:17:16.290
That's the case where
we have, a combination
00:17:16.290 --> 00:17:19.040
that gives the zero column.
00:17:19.040 --> 00:17:23.089
I'm often interested in the
case when my vectors are
00:17:23.089 --> 00:17:25.560
popped into a matrix.
00:17:25.560 --> 00:17:28.349
So the, the definition
over there of independence
00:17:28.349 --> 00:17:31.270
didn't talk about any matrix.
00:17:31.270 --> 00:17:36.130
The vectors didn't have to be
vectors in N dimensional space.
00:17:36.130 --> 00:17:38.530
And I want to give
you some examples
00:17:38.530 --> 00:17:41.710
of vectors that
aren't what you think
00:17:41.710 --> 00:17:43.970
of immediately as vectors.
00:17:43.970 --> 00:17:49.420
But most of the time, this is
-- the vectors we think of are
00:17:49.420 --> 00:17:51.200
columns.
00:17:51.200 --> 00:17:54.570
And we can put them in a matrix.
00:17:54.570 --> 00:17:57.330
And then independence
or dependence
00:17:57.330 --> 00:18:01.650
comes back to the null space.
00:18:01.650 --> 00:18:02.162
OK.
00:18:02.162 --> 00:18:03.620
So that's the idea
of independence.
00:18:06.410 --> 00:18:13.770
Can I just, yeah, let
me go on to spanning a
00:18:13.770 --> 00:18:19.480
What does it mean for a bunch
of vectors to span a space?
00:18:19.480 --> 00:18:20.580
space.
00:18:20.580 --> 00:18:23.800
Well, actually, we've
seen it already.
00:18:23.800 --> 00:18:28.040
You remember, if we had
a columns in a matrix,
00:18:28.040 --> 00:18:33.130
we took all their
combinations and that gave us
00:18:33.130 --> 00:18:36.640
the column space.
00:18:36.640 --> 00:18:41.250
Those vectors that we started
with span that column space.
00:18:41.250 --> 00:18:44.490
So spanning a space means --
00:18:44.490 --> 00:18:48.440
so let me move that
important stuff right up.
00:18:53.130 --> 00:18:54.050
OK.
00:18:54.050 --> 00:19:02.400
So vectors -- let me call
them, say, V1 up to --
00:19:02.400 --> 00:19:06.360
call you some different
letter, say Vl --
00:19:06.360 --> 00:19:15.680
span a space, a subspace,
or just a vector space
00:19:15.680 --> 00:19:24.790
I could say, span a
space means, means
00:19:24.790 --> 00:19:40.270
the space consists of all
combinations of those vectors.
00:19:46.464 --> 00:19:48.505
That's exactly what we
did with the column space.
00:19:51.040 --> 00:19:54.740
So now I could say in shorthand
the columns of a matrix
00:19:54.740 --> 00:19:57.090
span the column space.
00:19:57.090 --> 00:20:00.750
So you remember it's a bunch of
vectors that have this property
00:20:00.750 --> 00:20:05.450
that they span a space, and
actually if I give you a bunch
00:20:05.450 --> 00:20:06.480
of vectors and say --
00:20:06.480 --> 00:20:10.720
OK, let S be the
space that they span,
00:20:10.720 --> 00:20:14.870
in other words let S contain
all their combinations,
00:20:14.870 --> 00:20:17.750
that space S will
be the smallest
00:20:17.750 --> 00:20:21.830
space with those
vectors in it, right?
00:20:21.830 --> 00:20:24.300
Because any space with
those vectors in it
00:20:24.300 --> 00:20:28.730
must have all the combinations
of those vectors in it.
00:20:28.730 --> 00:20:34.540
And if I stop there, then
I've got the smallest space,
00:20:34.540 --> 00:20:37.300
and that's the space
that they span.
00:20:37.300 --> 00:20:37.800
OK.
00:20:37.800 --> 00:20:39.150
So I'm just --
00:20:39.150 --> 00:20:44.150
rather than, needing to say,
take all linear combinations
00:20:44.150 --> 00:20:48.950
and put them in a space,
I'm compressing that
00:20:48.950 --> 00:20:50.740
into the word span.
00:20:53.169 --> 00:20:53.835
Straightforward.
00:20:57.520 --> 00:21:00.015
So if I think of a, of
the column space of a
00:21:00.015 --> 00:21:00.515
OK. matrix.
00:21:03.890 --> 00:21:07.380
I've got their -- so I
start with the columns.
00:21:07.380 --> 00:21:08.930
I take all their combinations.
00:21:08.930 --> 00:21:10.760
That gives me the columns space.
00:21:10.760 --> 00:21:13.440
They span the column space.
00:21:13.440 --> 00:21:16.230
Now are they independent?
00:21:16.230 --> 00:21:19.620
Maybe yes, maybe no.
00:21:19.620 --> 00:21:23.060
It depends on the particular
columns that went into that
00:21:23.060 --> 00:21:24.470
matrix.
00:21:24.470 --> 00:21:29.700
But obviously I'm highly
interested in a set
00:21:29.700 --> 00:21:36.870
of vectors that spans a
space and is independent.
00:21:36.870 --> 00:21:41.630
That's, that means like I've
got the right number of vectors.
00:21:41.630 --> 00:21:47.580
If I didn't have all of them,
I wouldn't have my whole space.
00:21:47.580 --> 00:21:50.290
If I had more than that,
they probably wouldn't --
00:21:50.290 --> 00:21:52.450
they wouldn't be independent.
00:21:52.450 --> 00:21:56.860
So, like, basis -- and that's
the word that's coming --
00:21:56.860 --> 00:21:58.600
is just right.
00:21:58.600 --> 00:22:01.310
So here let me put
what that word means.
00:22:01.310 --> 00:22:14.270
A basis for a vector space is,
is a, is a sequence of vectors
00:22:14.270 --> 00:22:14.770
--
00:22:20.050 --> 00:22:27.040
shall I call them V1, V2,
up to let me say Vd now,
00:22:27.040 --> 00:22:33.600
I'll stop with that letters
-- that has two properties.
00:22:33.600 --> 00:22:36.230
I've got enough vectors
and not too many.
00:22:36.230 --> 00:22:39.340
It's a natural idea of a basis.
00:22:39.340 --> 00:22:42.570
So a basis is a bunch
of vectors in the space
00:22:42.570 --> 00:22:47.200
and it's a so it's a sequence
of vectors with two properties,
00:22:47.200 --> 00:22:50.050
with two properties.
00:22:54.760 --> 00:22:57.675
One, they are independent.
00:23:05.020 --> 00:23:07.990
And two -- you
know what's coming?
00:23:07.990 --> 00:23:10.040
-- they span the space.
00:23:20.560 --> 00:23:21.880
OK.
00:23:21.880 --> 00:23:25.050
Let me take --
00:23:25.050 --> 00:23:28.510
so time for examples, of course.
00:23:28.510 --> 00:23:32.320
So I'm asking you now
to put definition one,
00:23:32.320 --> 00:23:38.030
the definition of independence,
together with definition two,
00:23:38.030 --> 00:23:41.800
and let's look at examples,
because this is --
00:23:41.800 --> 00:23:44.840
this combination
means the set I've --
00:23:44.840 --> 00:23:47.690
of vectors I have is
just right, and the --
00:23:47.690 --> 00:23:51.210
so that this idea of a
basis will be central.
00:23:51.210 --> 00:23:54.480
I'll always be asking
you now for a basis.
00:23:54.480 --> 00:23:58.760
Whenever I look at a
subspace, if I ask you for --
00:23:58.760 --> 00:24:00.760
if you give me a basis
for that subspace,
00:24:00.760 --> 00:24:02.920
you've told me what it is.
00:24:02.920 --> 00:24:07.430
You've told me everything I need
to know about that subspace.
00:24:07.430 --> 00:24:10.520
Those -- I take their
combinations and I know that I
00:24:10.520 --> 00:24:12.460
need all the combinations.
00:24:12.460 --> 00:24:13.210
OK.
00:24:13.210 --> 00:24:13.790
Examples.
00:24:13.790 --> 00:24:16.640
OK, so examples of a basis.
00:24:16.640 --> 00:24:20.060
Let me start with two
dimensional space.
00:24:20.060 --> 00:24:22.350
Suppose the space
-- say example.
00:24:26.520 --> 00:24:31.150
The space is, oh,
let's make it R^3.
00:24:31.150 --> 00:24:35.980
Real three dimensional space.
00:24:35.980 --> 00:24:37.330
Give me one basis.
00:24:37.330 --> 00:24:38.705
One basis is?
00:24:43.970 --> 00:24:49.010
So I want some vectors, because
if I ask you for a basis,
00:24:49.010 --> 00:24:53.390
I'm asking you for vectors,
a little list of vectors.
00:24:53.390 --> 00:24:57.610
And it should be just right.
00:24:57.610 --> 00:25:02.880
So what would be a basis
for three dimensional space?
00:25:02.880 --> 00:25:05.540
Well, the first basis that
comes to mind, why don't we
00:25:05.540 --> 00:25:07.240
write that down.
00:25:07.240 --> 00:25:09.220
The first basis
that comes to mind
00:25:09.220 --> 00:25:18.850
is this vector, this
vector, and this vector.
00:25:18.850 --> 00:25:19.790
OK.
00:25:19.790 --> 00:25:23.540
That's one basis.
00:25:23.540 --> 00:25:27.190
Not the only basis, that's
going to be my point.
00:25:27.190 --> 00:25:30.930
But let's just see --
yes, that's a basis.
00:25:30.930 --> 00:25:33.500
Are, are those
vectors independent?
00:25:36.110 --> 00:25:40.120
So that's the like the x, y,
z axes, so if those are not
00:25:40.120 --> 00:25:41.780
independent, we're in trouble.
00:25:41.780 --> 00:25:43.580
Certainly, they are.
00:25:43.580 --> 00:25:48.630
Take a combination c1 of this
vector plus c2 of this vector
00:25:48.630 --> 00:25:51.340
plus c3 of that
vector and try to make
00:25:51.340 --> 00:25:55.290
it give the zero vector.
00:25:55.290 --> 00:25:57.110
What are the c-s?
00:25:57.110 --> 00:26:03.080
If c1 of that plus c2 of that
plus c3 of that gives me 0 0 0,
00:26:03.080 --> 00:26:05.600
then the c-s are all --
00:26:05.600 --> 00:26:06.920
0, right.
00:26:06.920 --> 00:26:09.980
So that's the test
for independence.
00:26:09.980 --> 00:26:16.210
In the language of matrices,
which was under that board,
00:26:16.210 --> 00:26:19.700
I could make those the
columns of a matrix.
00:26:19.700 --> 00:26:22.880
Well, it would be
the identity matrix.
00:26:22.880 --> 00:26:25.590
Then I would ask, what's the
null space of the identity
00:26:25.590 --> 00:26:26.760
matrix?
00:26:26.760 --> 00:26:30.560
And you would say it's
only the zero vector.
00:26:30.560 --> 00:26:34.740
And I would say, fine, then
the columns are independent.
00:26:34.740 --> 00:26:38.370
The only thing -- the identity
times a vector giving zero,
00:26:38.370 --> 00:26:40.810
the only vector that
does that is zero.
00:26:40.810 --> 00:26:41.980
OK.
00:26:41.980 --> 00:26:45.710
Now that's not the only basis.
00:26:45.710 --> 00:26:46.660
Far from it.
00:26:46.660 --> 00:26:50.680
Tell me another basis, a
second basis, another basis.
00:26:57.560 --> 00:27:03.030
So, give me -- well,
I'll just start it out.
00:27:03.030 --> 00:27:04.090
One one two.
00:27:06.950 --> 00:27:10.780
Two two five.
00:27:10.780 --> 00:27:14.420
Suppose I stopped there.
00:27:14.420 --> 00:27:20.980
Has that little bunch of
vectors got the properties that
00:27:20.980 --> 00:27:24.350
I'm asking for in
a basis for R^3?
00:27:24.350 --> 00:27:26.880
We're looking for
a basis for R^3.
00:27:26.880 --> 00:27:30.430
Are they independent,
those two column vectors?
00:27:30.430 --> 00:27:31.330
Yes.
00:27:31.330 --> 00:27:33.610
Do they span R^3?
00:27:33.610 --> 00:27:34.340
No.
00:27:34.340 --> 00:27:36.440
Our feeling is no.
00:27:36.440 --> 00:27:37.550
Our feeling is no.
00:27:37.550 --> 00:27:41.640
Our feeling is that there're
some vectors in R3 that
00:27:41.640 --> 00:27:44.120
are not combinations of those.
00:27:44.120 --> 00:27:44.930
OK.
00:27:44.930 --> 00:27:47.830
So suppose I add in --
00:27:47.830 --> 00:27:50.390
I need another vector
then, because these two
00:27:50.390 --> 00:27:52.021
don't span the space.
00:27:52.021 --> 00:27:52.520
OK.
00:27:52.520 --> 00:27:56.360
Now it would be foolish for me
to put in three three seven,
00:27:56.360 --> 00:27:58.810
right, as the third vector.
00:27:58.810 --> 00:27:59.950
That would be a goof.
00:27:59.950 --> 00:28:03.380
Because that, if I put
in three three seven,
00:28:03.380 --> 00:28:07.680
those vectors would
be dependent, right?
00:28:07.680 --> 00:28:09.440
If I put in three
three seven, it
00:28:09.440 --> 00:28:12.170
would be the sum
of those two, it
00:28:12.170 --> 00:28:15.020
would lie in the
same plane as those.
00:28:15.020 --> 00:28:18.590
It wouldn't be independent.
00:28:18.590 --> 00:28:21.930
My attempt to create
a basis would be dead.
00:28:21.930 --> 00:28:26.420
But if I take -- so
what vector can I take?
00:28:26.420 --> 00:28:30.340
I can take any vector
that's not in that plane.
00:28:30.340 --> 00:28:31.430
Let me try --
00:28:31.430 --> 00:28:33.590
I hope that 3 3 8 would do it.
00:28:37.340 --> 00:28:40.470
At least it's not the
sum of those two vectors.
00:28:40.470 --> 00:28:44.160
But I believe that's a basis.
00:28:44.160 --> 00:28:49.310
And what's the test then,
for that to be a basis?
00:28:49.310 --> 00:28:53.160
Because I just picked those
numbers, and if I had picked,
00:28:53.160 --> 00:29:02.300
5 7 -14 how would we know do
we have a basis or don't we?
00:29:02.300 --> 00:29:05.430
You would put them in
the columns of a matrix,
00:29:05.430 --> 00:29:09.280
and you would do
elimination, row reduction --
00:29:09.280 --> 00:29:15.340
and you would see do you
get any free variables
00:29:15.340 --> 00:29:18.500
or are all the
columns pivot columns.
00:29:18.500 --> 00:29:20.590
Well now actually
we have a square --
00:29:20.590 --> 00:29:22.760
the matrix would
be three by three.
00:29:22.760 --> 00:29:26.260
So, what's the test
on the matrix then?
00:29:29.280 --> 00:29:34.790
The matrix -- so in this case,
when my space is R^3 and I have
00:29:34.790 --> 00:29:44.740
three vectors, my matrix is
square and what I asking about
00:29:44.740 --> 00:29:49.780
that matrix in order for
those columns to be a basis?
00:29:49.780 --> 00:29:51.190
So in this --
00:29:51.190 --> 00:30:06.380
for R^n, if I have -- n vectors
give a basis if the n by n
00:30:06.380 --> 00:30:19.850
matrix with those columns,
with those columns, is what?
00:30:19.850 --> 00:30:24.070
What's the requirement
on that matrix?
00:30:24.070 --> 00:30:27.010
Invertible, right, right.
00:30:27.010 --> 00:30:28.590
The matrix should be invertible.
00:30:28.590 --> 00:30:33.187
For a square matrix, that's
the, that's the perfect answer.
00:30:33.187 --> 00:30:33.770
Is invertible.
00:30:38.260 --> 00:30:43.380
So that's when, that's when the
space is the whole space R^n.
00:30:46.140 --> 00:30:50.360
Let me, let me be sure
you're with me here.
00:30:50.360 --> 00:30:53.540
Let me remove that.
00:30:53.540 --> 00:31:01.580
Are those two vectors a
basis for any space at all?
00:31:01.580 --> 00:31:04.020
Is there a vector
space that those really
00:31:04.020 --> 00:31:08.540
are a basis for, those, that
pair of vectors, this guy
00:31:08.540 --> 00:31:11.580
and this 1, 1 1 2 and 2 2 5?
00:31:11.580 --> 00:31:15.360
Is there a space for
which that's a basis?
00:31:15.360 --> 00:31:16.220
Sure.
00:31:16.220 --> 00:31:21.970
They're independent, so they
satisfy the first requirement,
00:31:21.970 --> 00:31:24.740
so what space shall I take
for them to be a basis
00:31:24.740 --> 00:31:25.240
of?
00:31:25.240 --> 00:31:29.080
What spaces will
they be a basis for?
00:31:29.080 --> 00:31:31.180
The one they span.
00:31:31.180 --> 00:31:32.350
Their combinations.
00:31:32.350 --> 00:31:33.920
It's a plane, right?
00:31:33.920 --> 00:31:36.340
It'll be a plane inside R^3.
00:31:36.340 --> 00:31:40.930
So if I take this vector
1 1 2, say it goes there,
00:31:40.930 --> 00:31:44.240
and this vector 2 2
5, say it goes there,
00:31:44.240 --> 00:31:50.797
those are a basis for --
because they span a plane.
00:31:50.797 --> 00:31:52.880
And they're a basis for
the plane, because they're
00:31:52.880 --> 00:31:53.550
independent.
00:31:53.550 --> 00:31:56.800
If I stick in some
third guy, like 3 3 7,
00:31:56.800 --> 00:32:01.050
which is in the plane -- suppose
I put in, try to put in 3 3 7,
00:32:01.050 --> 00:32:05.740
then the three vectors
would still span the plane,
00:32:05.740 --> 00:32:09.180
but they wouldn't be a basis
anymore because they're not
00:32:09.180 --> 00:32:12.260
independent anymore.
00:32:12.260 --> 00:32:19.420
So, we're looking at
the question of --
00:32:19.420 --> 00:32:21.360
again,
00:32:21.360 --> 00:32:25.390
OK. the case with
independent columns
00:32:25.390 --> 00:32:32.090
is the case where the column
vectors span the column space.
00:32:32.090 --> 00:32:35.830
They're independent, so they're
a basis for the column space.
00:32:35.830 --> 00:32:36.520
OK.
00:32:36.520 --> 00:32:42.490
So now there's one
bit of intuition.
00:32:42.490 --> 00:32:46.420
Let me go back to all of R^n.
00:32:46.420 --> 00:32:48.150
So I -- where I put 3 3 8.
00:32:51.110 --> 00:32:51.740
OK.
00:32:51.740 --> 00:32:56.510
The first message is that the
basis is not unique, right.
00:32:56.510 --> 00:32:58.090
There's zillions of bases.
00:32:58.090 --> 00:33:02.790
I take any invertible
three by three matrix,
00:33:02.790 --> 00:33:07.800
its columns are a basis for R^3.
00:33:07.800 --> 00:33:11.200
The column space is
R^3, and if those,
00:33:11.200 --> 00:33:15.180
if that matrix is invertible,
those columns are independent,
00:33:15.180 --> 00:33:17.100
I've got a basis for R^3.
00:33:17.100 --> 00:33:19.860
So there're many, many bases.
00:33:19.860 --> 00:33:27.730
But there is something in
common for all those bases.
00:33:27.730 --> 00:33:32.810
There's something that this
basis shares with that basis
00:33:32.810 --> 00:33:35.880
and every other basis for R^3.
00:33:35.880 --> 00:33:37.515
And what's that?
00:33:40.040 --> 00:33:47.410
Well, you saw it coming, because
when I stopped here and asked
00:33:47.410 --> 00:33:51.180
if that was a basis
for R^3, you said no.
00:33:51.180 --> 00:33:54.730
And I know that you said
no because you knew there
00:33:54.730 --> 00:33:57.410
weren't enough vectors there.
00:33:57.410 --> 00:34:03.670
And the great fact is that
there're many, many bases,
00:34:03.670 --> 00:34:12.139
but -- let me put in somebody
else, just for variety.
00:34:12.139 --> 00:34:14.850
There are many, many
bases, but they all
00:34:14.850 --> 00:34:18.639
have the same number of vectors.
00:34:18.639 --> 00:34:21.350
If we're talking
about the space R^3,
00:34:21.350 --> 00:34:25.040
then that number of
vectors is three.
00:34:25.040 --> 00:34:27.650
If we're talking
about the space R^n,
00:34:27.650 --> 00:34:31.320
then that number
of vectors is n.
00:34:31.320 --> 00:34:36.389
If we're talking about
some other space,
00:34:36.389 --> 00:34:41.570
the column space of some matrix,
or the null space of some
00:34:41.570 --> 00:34:45.420
matrix, or some other space
that we haven't even thought
00:34:45.420 --> 00:34:52.820
of, then that still is
true that every basis --
00:34:52.820 --> 00:34:57.270
that there're lots of bases but
every basis has the same number
00:34:57.270 --> 00:34:57.950
of vectors.
00:34:57.950 --> 00:35:02.000
Let me write that
great fact down.
00:35:02.000 --> 00:35:07.940
Every basis --
we're given a space.
00:35:07.940 --> 00:35:10.680
Given a space.
00:35:13.950 --> 00:35:18.840
R^3 or R^n or some other column
space of a matrix or the null
00:35:18.840 --> 00:35:21.790
space of a matrix or
some other vector space.
00:35:21.790 --> 00:35:25.200
Then the great fact
is that every basis
00:35:25.200 --> 00:35:40.365
for this, for the space has
the same number of vectors.
00:35:47.750 --> 00:35:50.770
If one basis has six vectors,
then every other basis
00:35:50.770 --> 00:35:52.770
has six vectors.
00:35:52.770 --> 00:35:56.050
So that number six
is telling me like
00:35:56.050 --> 00:35:59.250
it's telling me how
big is the space.
00:35:59.250 --> 00:36:01.640
It's telling me
how many vectors do
00:36:01.640 --> 00:36:04.650
I have to have to have a basis.
00:36:04.650 --> 00:36:08.440
And of course we're
seeing it this way.
00:36:08.440 --> 00:36:10.900
That number six, if
we had seven vectors,
00:36:10.900 --> 00:36:13.150
then we've got too many.
00:36:13.150 --> 00:36:17.070
If we have five vectors
we haven't got enough.
00:36:17.070 --> 00:36:21.860
Sixes are like just right
for whatever space that is.
00:36:21.860 --> 00:36:24.600
And what do we call that number?
00:36:24.600 --> 00:36:29.550
That number is -- now I'm ready
for the last definition today.
00:36:29.550 --> 00:36:33.650
It's the dimension
of that space.
00:36:33.650 --> 00:36:37.410
So every basis for a space has
the same number of vectors in
00:36:37.410 --> 00:36:37.910
it.
00:36:37.910 --> 00:36:41.860
Not the same vectors,
all sorts of bases --
00:36:41.860 --> 00:36:44.850
but the same number of
vectors is always the same,
00:36:44.850 --> 00:36:47.010
and that number
is the dimension.
00:36:47.010 --> 00:36:47.900
This is definitional.
00:36:50.780 --> 00:36:58.245
This number is the
dimension of the space.
00:37:03.390 --> 00:37:03.890
OK.
00:37:06.670 --> 00:37:08.417
OK.
00:37:08.417 --> 00:37:09.375
Let's do some examples.
00:37:12.930 --> 00:37:14.410
Because now we've
got definitions.
00:37:14.410 --> 00:37:17.550
Let me repeat the four
things, the four words that
00:37:17.550 --> 00:37:19.270
have now got defined.
00:37:19.270 --> 00:37:23.320
Independence, that looks
at combinations not
00:37:23.320 --> 00:37:24.560
being zero.
00:37:24.560 --> 00:37:27.700
Spanning, that looks at
all the combinations.
00:37:27.700 --> 00:37:30.580
Basis, that's the
one that combines
00:37:30.580 --> 00:37:32.360
independence and spanning.
00:37:32.360 --> 00:37:36.460
And now we've got the idea
of the dimension of a space.
00:37:36.460 --> 00:37:40.470
It's the number of
vectors in any basis,
00:37:40.470 --> 00:37:43.160
because all bases
have the same number.
00:37:43.160 --> 00:37:44.580
OK.
00:37:44.580 --> 00:37:47.750
Let's take examples.
00:37:47.750 --> 00:37:53.790
Suppose I take, my space
is -- examples now --
00:37:53.790 --> 00:37:58.170
space is the, say, the
column space of this matrix.
00:37:58.170 --> 00:38:00.240
Let me write down a matrix.
00:38:00.240 --> 00:38:06.370
1 1 1, 2 1 2, and I'll
-- just to make it clear,
00:38:06.370 --> 00:38:12.430
I'll take the sum there, 3 2 3,
and let me take the sum of all
00:38:12.430 --> 00:38:15.050
-- oh, let me put
in one -- yeah,
00:38:15.050 --> 00:38:18.530
I'll put in one one one again.
00:38:18.530 --> 00:38:20.300
OK.
00:38:20.300 --> 00:38:21.260
So that's four vectors.
00:38:24.710 --> 00:38:29.310
OK, do they span the column
space of that matrix?
00:38:29.310 --> 00:38:35.290
Let me repeat, do they span the
column space of that matrix?
00:38:35.290 --> 00:38:37.700
By definition, that's
what the column space --
00:38:37.700 --> 00:38:39.280
Yes. where it comes from.
00:38:39.280 --> 00:38:41.470
Are they a basis for
the column space?
00:38:41.470 --> 00:38:43.920
Are they independent?
00:38:43.920 --> 00:38:45.860
No, they're not independent.
00:38:45.860 --> 00:38:49.460
There's something
in that null space.
00:38:49.460 --> 00:38:55.140
Maybe we can -- so let's look
at the null space of the matrix.
00:38:55.140 --> 00:38:58.490
Tell me a vector that's in
the null space of that matrix.
00:39:01.090 --> 00:39:05.320
So I'm looking for some vector
that combines those columns
00:39:05.320 --> 00:39:08.350
and produces the zero column.
00:39:08.350 --> 00:39:10.680
Or in other words, I'm
looking for solutions
00:39:10.680 --> 00:39:12.280
to A X equals zero.
00:39:12.280 --> 00:39:16.930
So tell me a vector
in the null space.
00:39:16.930 --> 00:39:21.070
Maybe -- well, this was, this
column was that one plus that
00:39:21.070 --> 00:39:25.030
one, so maybe if I have one of
those and minus one of those
00:39:25.030 --> 00:39:26.560
that would be a
vector in the null
00:39:26.560 --> 00:39:27.060
space.
00:39:29.450 --> 00:39:33.440
So, you've already told me now,
are those vectors independent,
00:39:33.440 --> 00:39:38.140
the answer is -- those column
vectors, the answer is --
00:39:38.140 --> 00:39:38.670
no.
00:39:38.670 --> 00:39:39.170
Right?
00:39:39.170 --> 00:39:40.770
They're not independent.
00:39:40.770 --> 00:39:44.060
Because -- you knew they
weren't independent.
00:39:44.060 --> 00:39:47.060
Anyway, minus one
of this minus one
00:39:47.060 --> 00:39:50.530
of this plus one of this zero
of that is the zero vector.
00:39:54.120 --> 00:39:55.660
OK, so they're not independent.
00:39:55.660 --> 00:39:55.720
OK.
00:39:55.720 --> 00:39:57.500
They span, but they're
not independent.
00:39:57.500 --> 00:40:04.560
Tell me a basis for
that column space.
00:40:04.560 --> 00:40:06.300
What's a basis for
the column space?
00:40:06.300 --> 00:40:09.220
These are all the questions that
the homework asks, the quizzes
00:40:09.220 --> 00:40:11.800
ask, the final exam will ask.
00:40:11.800 --> 00:40:17.710
Find a basis for the column
space of this matrix.
00:40:17.710 --> 00:40:20.280
OK.
00:40:20.280 --> 00:40:22.500
Now there's many
answers, but give me
00:40:22.500 --> 00:40:25.590
the most natural answer.
00:40:25.590 --> 00:40:29.670
Columns one and two.
00:40:29.670 --> 00:40:31.170
Columns one and two.
00:40:31.170 --> 00:40:32.410
That's the natural answer.
00:40:32.410 --> 00:40:35.320
Those are the pivot
columns, because, I mean,
00:40:35.320 --> 00:40:37.140
we s- we begin systematically.
00:40:37.140 --> 00:40:39.300
We look at the first
column, it's OK.
00:40:39.300 --> 00:40:41.330
We can put that in the basis.
00:40:41.330 --> 00:40:43.550
We look at the second
column, it's OK.
00:40:43.550 --> 00:40:46.300
We can put that in the basis.
00:40:46.300 --> 00:40:48.960
The third column we
can't put in the basis.
00:40:48.960 --> 00:40:54.070
The fourth column
we can't, again.
00:40:54.070 --> 00:40:57.570
So the rank of the matrix is --
00:40:57.570 --> 00:40:59.850
what's the rank of our matrix?
00:40:59.850 --> 00:41:01.210
Two.
00:41:01.210 --> 00:41:02.000
Two.
00:41:02.000 --> 00:41:06.610
And, and now that rank is also
-- we also have another word.
00:41:06.610 --> 00:41:08.710
We, we have a
great theorem here.
00:41:08.710 --> 00:41:23.060
The rank of A, that rank r,
is the number of pivot columns
00:41:23.060 --> 00:41:25.340
and it's also --
00:41:25.340 --> 00:41:27.890
well, so now please
use my new word.
00:41:32.640 --> 00:41:34.700
This, it's the number
two, of course,
00:41:34.700 --> 00:41:41.790
two is the rank
of my matrix, it's
00:41:41.790 --> 00:41:44.440
the number of pivot columns,
those pivot columns form
00:41:44.440 --> 00:41:49.104
a basis, of course,
so what's two?
00:41:49.104 --> 00:41:49.895
It's the dimension.
00:41:52.820 --> 00:41:55.580
The rank of A, the
number of pivot columns,
00:41:55.580 --> 00:42:02.555
is the dimension of
the column space.
00:42:07.450 --> 00:42:08.840
Of course, you say.
00:42:08.840 --> 00:42:10.670
It had to be.
00:42:10.670 --> 00:42:12.000
Right.
00:42:12.000 --> 00:42:17.440
But just watch, look
for one moment at the,
00:42:17.440 --> 00:42:20.240
the language, the
way the English words
00:42:20.240 --> 00:42:21.810
get involved here.
00:42:21.810 --> 00:42:28.610
I take the rank of a matrix,
the rank of a matrix.
00:42:28.610 --> 00:42:34.580
It's a number of columns
and it's the dimension of --
00:42:34.580 --> 00:42:37.810
not the dimension of the matrix,
that's what I want to say.
00:42:37.810 --> 00:42:43.890
It's the dimension of a space,
a subspace, the column space.
00:42:43.890 --> 00:42:46.800
Do you see, I don't
take the dimension of A.
00:42:46.800 --> 00:42:49.600
That's not what I want.
00:42:49.600 --> 00:42:53.190
I'm looking for the dimension
of the column space of A.
00:42:53.190 --> 00:42:56.050
If you use those words right,
it shows you've got the idea
00:42:56.050 --> 00:42:57.400
right.
00:42:57.400 --> 00:42:59.170
Similarly here.
00:42:59.170 --> 00:43:03.430
I don't talk about the
rank of a subspace.
00:43:03.430 --> 00:43:05.440
It's a matrix that has a rank.
00:43:05.440 --> 00:43:08.040
I talk about the
rank of a matrix.
00:43:08.040 --> 00:43:11.750
And the beauty is that
these definitions just
00:43:11.750 --> 00:43:14.160
merge so that the
rank of a matrix
00:43:14.160 --> 00:43:16.170
is the dimension of
its column space.
00:43:16.170 --> 00:43:18.840
And in this example it's two.
00:43:18.840 --> 00:43:22.570
And then the further
question is, what's a basis?
00:43:22.570 --> 00:43:25.450
And the first two
columns are a basis.
00:43:25.450 --> 00:43:27.500
Tell me another basis.
00:43:27.500 --> 00:43:29.710
Another basis for
the columns space.
00:43:29.710 --> 00:43:31.390
You see I just keep
hammering away.
00:43:31.390 --> 00:43:35.140
I apologize, but it's,
I have to be sure you
00:43:35.140 --> 00:43:36.660
have the idea of basis.
00:43:36.660 --> 00:43:40.940
Tell me another basis
for the column space.
00:43:40.940 --> 00:43:47.410
Well, you could take
columns one and three.
00:43:47.410 --> 00:43:49.930
That would be a basis
for the column space.
00:43:49.930 --> 00:43:53.220
Or columns two and
three would be a basis.
00:43:53.220 --> 00:43:55.660
Or columns two and four.
00:43:55.660 --> 00:43:58.871
Or tell me another basis that's
not made out of those columns
00:43:58.871 --> 00:43:59.370
at all?
00:44:03.290 --> 00:44:07.180
So -- I guess I'm giving you
infinitely many possibilities,
00:44:07.180 --> 00:44:11.150
so I can't expect a
unanimous answer here.
00:44:11.150 --> 00:44:14.770
I'll tell you -- but let's
look at another basis, though.
00:44:14.770 --> 00:44:17.836
I'll just -- because it's
only one out of zillions,
00:44:17.836 --> 00:44:19.960
I'm going to put it down
and I'm going to erase it.
00:44:19.960 --> 00:44:26.680
Another basis for the
column space would be --
00:44:26.680 --> 00:44:27.879
let's see.
00:44:27.879 --> 00:44:29.670
I'll put in some things
that are not there.
00:44:29.670 --> 00:44:33.830
Say, oh well, just to make
it -- my life easy, 2 2 2.
00:44:33.830 --> 00:44:37.040
That's in the column space.
00:44:37.040 --> 00:44:40.380
And, that was sort of obvious.
00:44:40.380 --> 00:44:43.458
Let me take the sum
of those, say 6 4 6.
00:44:46.810 --> 00:44:52.040
Or the sum of all of the
columns, 7 5 7, why not.
00:44:52.040 --> 00:44:54.910
That's in the column space.
00:44:54.910 --> 00:44:59.540
Those are independent and
I've got the number right,
00:44:59.540 --> 00:45:01.290
I've got two.
00:45:01.290 --> 00:45:02.940
Actually, this is a key point.
00:45:05.890 --> 00:45:09.170
If you know the dimension of
the space you're working with,
00:45:09.170 --> 00:45:14.480
and we know that this column
-- we know that the dimension,
00:45:14.480 --> 00:45:19.430
DIM, the dimension of
the column space is two.
00:45:23.730 --> 00:45:30.120
If you know the
dimension, then --
00:45:30.120 --> 00:45:33.670
and we have a couple of
vectors that are independent,
00:45:33.670 --> 00:45:35.710
they'll automatically be a
00:45:35.710 --> 00:45:36.550
basis.
00:45:36.550 --> 00:45:38.890
If we've got the number
of vectors right,
00:45:38.890 --> 00:45:43.760
two vectors in this case,
then if they're independent,
00:45:43.760 --> 00:45:47.037
they can't help
but span the space.
00:45:47.037 --> 00:45:48.620
Because if they
didn't span the space,
00:45:48.620 --> 00:45:52.130
there'd be a third guy
to help span the space,
00:45:52.130 --> 00:45:54.160
but it couldn't be independent.
00:45:54.160 --> 00:45:58.190
So, it just has
to be independent
00:45:58.190 --> 00:46:01.640
if we've got the numbers right.
00:46:01.640 --> 00:46:02.590
And they span.
00:46:02.590 --> 00:46:03.280
OK.
00:46:03.280 --> 00:46:04.100
Very good.
00:46:04.100 --> 00:46:06.060
So you got the
dimension of a space.
00:46:06.060 --> 00:46:08.870
So this was another basis
that I just invented.
00:46:08.870 --> 00:46:09.470
OK.
00:46:09.470 --> 00:46:16.270
Now, now I get to ask
about the null space.
00:46:16.270 --> 00:46:18.320
What's the dimension
of the null space?
00:46:18.320 --> 00:46:20.700
So we, we got a
great fact there,
00:46:20.700 --> 00:46:28.560
the dimension of the
column space is the rank.
00:46:28.560 --> 00:46:30.760
Now I want to ask you
about the null space.
00:46:30.760 --> 00:46:34.190
That's the other
part of the lecture,
00:46:34.190 --> 00:46:38.320
and it'll go on to
the next lecture.
00:46:38.320 --> 00:46:40.320
OK.
00:46:40.320 --> 00:46:44.230
So we know the dimension of the
column space is two, the rank.
00:46:44.230 --> 00:46:46.280
What about the null space?
00:46:46.280 --> 00:46:47.870
This is a vector
in the null space.
00:46:47.870 --> 00:46:49.770
Are there other vectors
in the null space?
00:46:52.280 --> 00:46:53.530
Yes or no?
00:46:53.530 --> 00:46:54.870
Yes.
00:46:54.870 --> 00:46:58.820
So this isn't a basis because
it's doesn't span, right?
00:46:58.820 --> 00:47:01.970
There's more in the null
space than we've got so far.
00:47:01.970 --> 00:47:04.570
I need another vector at least.
00:47:04.570 --> 00:47:08.790
So tell me another
vector in the null space.
00:47:08.790 --> 00:47:12.290
Well, the natural choice, the
choice you naturally think of
00:47:12.290 --> 00:47:16.140
is I'm going on to
the fourth column,
00:47:16.140 --> 00:47:20.210
I'm letting that free
variable be a one,
00:47:20.210 --> 00:47:23.750
and that free variable
be a zero, and I'm asking
00:47:23.750 --> 00:47:26.130
is that fourth
column a combination
00:47:26.130 --> 00:47:27.410
of my pivot columns?
00:47:27.410 --> 00:47:28.670
Yes, it is.
00:47:28.670 --> 00:47:31.650
And it's -- that will do.
00:47:35.730 --> 00:47:37.820
So what I've written
there are actually the two
00:47:37.820 --> 00:47:39.860
special solutions, right?
00:47:39.860 --> 00:47:44.520
I took the two free
variables, free and free.
00:47:44.520 --> 00:47:50.410
I gave them the values 1 0
or 0 I figured out the rest.
00:47:50.410 --> 00:47:53.640
So do you see, let me
just say it in words.
00:47:53.640 --> 00:47:58.000
This vector, these vectors in
the null space are telling me,
00:47:58.000 --> 00:48:00.190
they're telling me
the combinations
00:48:00.190 --> 00:48:02.610
of the columns that give zero.
00:48:02.610 --> 00:48:08.770
They're telling me in what way
the, the columns are dependent.
00:48:08.770 --> 00:48:10.770
That's what the
null space is doing.
00:48:10.770 --> 00:48:13.890
Have I got enough now?
00:48:13.890 --> 00:48:15.780
And what's the null space now?
00:48:15.780 --> 00:48:18.540
We have to think
about the null space.
00:48:18.540 --> 00:48:20.740
These are two vectors
in the null space.
00:48:20.740 --> 00:48:21.790
They're independent.
00:48:21.790 --> 00:48:24.620
Are they a basis
for the null space?
00:48:24.620 --> 00:48:27.650
What's the dimension
of the null space?
00:48:27.650 --> 00:48:30.100
You see that those questions
just keep coming up all the
00:48:30.100 --> 00:48:31.120
time.
00:48:31.120 --> 00:48:34.500
Are they a basis
for the null space?
00:48:34.500 --> 00:48:36.940
You can tell me the answer
even though we haven't
00:48:36.940 --> 00:48:39.460
written out a proof of that.
00:48:39.460 --> 00:48:40.190
Can you?
00:48:40.190 --> 00:48:41.050
Yes or no?
00:48:41.050 --> 00:48:44.510
Do these two special
solutions form
00:48:44.510 --> 00:48:46.400
a basis for the null space?
00:48:46.400 --> 00:48:48.790
In other words,
does the null space
00:48:48.790 --> 00:48:52.390
consist of all combinations
of those two guys?
00:48:52.390 --> 00:48:54.110
Yes or no?
00:48:54.110 --> 00:48:55.180
Yes.
00:48:55.180 --> 00:48:56.880
Yes.
00:48:56.880 --> 00:48:58.970
The null space is
two dimensional.
00:48:58.970 --> 00:49:01.500
The null space, the
dimension of the null space,
00:49:01.500 --> 00:49:03.530
is the number of free variables.
00:49:03.530 --> 00:49:09.330
So the dimension
of the null space
00:49:09.330 --> 00:49:12.065
is the number of free variables.
00:49:17.260 --> 00:49:21.420
And at the last second,
give me the formula.
00:49:21.420 --> 00:49:24.490
This is then the key
formula that we know.
00:49:24.490 --> 00:49:29.740
How many free variables are
there in terms of R, the rank,
00:49:29.740 --> 00:49:35.020
m -- the number of rows,
n, the number of columns?
00:49:35.020 --> 00:49:36.740
What do we get?
00:49:36.740 --> 00:49:42.360
We have n columns, r of
them are pivot columns,
00:49:42.360 --> 00:49:48.200
so n-r is the number of free
columns, free variables.
00:49:48.200 --> 00:49:51.950
And now it's the dimension
of the null space.
00:49:51.950 --> 00:49:53.000
OK.
00:49:53.000 --> 00:49:53.950
That's great.
00:49:53.950 --> 00:49:57.280
That's the key spaces, their
bases, and their dimensions.
00:49:57.280 --> 00:49:58.830
Thanks.