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OK, when the camera says,
we'll start.

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00:00:05 --> 00:00:08.93
You want to give me a signal?

3
00:00:08.93 --> 00:00:14
OK, this is lecture eight in
linear algebra,

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and this is the lecture where
we completely solve linear

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equations.

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So Ax=b.

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That's our goal.

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If it has a solution.

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It certainly can happen that
there is no solution.

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00:00:34 --> 00:00:38.86
We have to identify that
possibility by elimination.

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And then if there is a solution
we want to find out is there

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only one solution or are -- is
there a whole family of

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solutions, and then find them
all.

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OK.

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Can I use as an example the
same matrix that I had last time

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when we were looking for the
null space.

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So the, the matrix has rows 1 2
2 2, 2 4 6 8,

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00:01:08 --> 00:01:15
and the third row -- you
remember the main point was the

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third row, 3 6 8 10,
is the sum of row one plus row

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two.

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In other words,
if I add those left-hand sides,

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I get the third left-hand side.

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So you can tell me right away
what elimination is going to

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discover about the right-hand
sides.

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What's -- there is a condition
on b1, b2, and b3 for this

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system to have a solution.

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Most cases -- if I took these
numbers to be one --

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5, and 17, there would not be a
solution.

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In fact, if I took those first
numbers to be 1 and 5,

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what is the only b3 that would
be OK?

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Six.

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If the left-hand -- if these
left-hand sides add up to that,

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then B --
I need b1 plus b2 to equal b3.

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Let's just see how elimination
discovers that.

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But we can see it coming,
right?

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That if -- let me say it in
other words.

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If some combination on the
left-hand side gives all 0s then

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the same combination on the
right-hand side must give 0.

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OK.

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So let me take that example and
write down instead of copying

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out all the plus signs,
let me write down the matrix.

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1 2 2 2, 2 4 6 8,
and that 6 3 8 10,

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where the third row is the sum
of the first two rows.

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00:02:54 --> 00:02:58
Now how do we deal with the
right-hand side?

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00:02:58 --> 00:03:03
That's -- we want to do the
same thing to the right-hand

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side that we're doing to these
rows on the left side,

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so we just tack on the
right-hand side as another

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vector, another column.

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So this is the augmented
matrix.

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It's, it's the matrix A with
the vector b tacked on.

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In Matlab, that's all you would
need to type.

52
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OK.

53
00:03:33 --> 00:03:36
So we do elimination on that.

54
00:03:36 --> 00:03:42
Can we just do elimination
quickly?

55
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The first pivot is fine,
I subtract two of this away

56
00:03:48 --> 00:03:55
from this, three of this away
from this, so I have 1 2 2 2 b1.

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00:03:55 --> 00:04:01.7
Two of those away will give me
0 0 2 and 4, and that was b2

58
00:04:01.7 --> 00:04:03
minus two b1.

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00:04:03 --> 00:04:08
I, I have to do the same thing
to that third,

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00:04:08 --> 00:04:11
that last column.

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00:04:11 --> 00:04:16
And then three of these away
from this gave me 0 0 2 4 b3

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minus three b1s.

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So that's the,
that's elimination with the

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first column completed.

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We move on.

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00:04:24 --> 00:04:27
There's the first pivot still.

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Here is the second pivot.

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We're always remembering,
now, these are then going to be

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the pivot columns.

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00:04:37 --> 00:04:44
And let me get the final result
-- well, let me -- can I do it

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by eraser?

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00:04:45 --> 00:04:52
We're capable of subtracting
this row from this row,

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00:04:52 --> 00:04:59
just by -- that'll knock this
out completely and give me the

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row of 0s, and on the right-hand
side, when I subtract this away

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from this, what do I have?

76
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I think I have b3 minus a b2,
and I had minus three b1s.

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00:05:18 --> 00:05:25
This is going to,
it's going to be a minus a b1.

78
00:05:25 --> 00:05:29
Oh yeah that's exactly what I
expect.

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00:05:29 --> 00:05:35.68
So now the -- what's the last
equation?

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00:05:35.68 --> 00:05:38
The last equation,
this represented by this zero

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00:05:38 --> 00:05:42
row, that last equation is,
says 0 equals b3 minus b2 minus

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00:05:42 --> 00:05:42
b1.

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00:05:42 --> 00:05:45
So that's the condition for
solvability.

84
00:05:45 --> 00:05:48.58
That's the condition on the
right-hand side that we

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expected.

86
00:05:49 --> 00:05:53
It says that b1+b2 has to match
b3, and if our numbers happen to

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have been 1, 5,
and 6 -- so let me take,

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suppose b is 1 5 6.

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00:05:58 --> 00:06:00
That's an OK b.

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00:06:00 --> 00:06:08
And when I do this elimination,
what will I have?

91
00:06:08 --> 00:06:15
The b1 will still be a 1.
b2 would be 5 minus 2,

92
00:06:15 --> 00:06:22
this would be a 3.
5 -- my 6 minus 5 minus 1,

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this will be -- this is the
main point -- this will be a 0,

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thanks.

95
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OK.

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00:06:28 --> 00:06:31
So the last equation is OK now.

97
00:06:31 --> 00:06:37
And I can proceed to solve the
two equations that are really

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there with four unknowns.

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00:06:40 --> 00:06:45
OK, I, I, I want to do that,
so this, this b is OK.

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It allows a solution.

101
00:06:48 --> 00:06:54
We're going to be,
naturally, interested to keep

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track what are the conditions on
b that make the equation

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00:07:01 --> 00:07:02.21
solvable.

104
00:07:02.21 --> 00:07:09
So let me write down what we
already see before I continue to

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solve it.

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Let me first -- solvability,
solvability.

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So which -- so this is the
condition on the right-hand

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sides.

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And what is that condition?

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This is solvability always of
Ax=b.

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So Ax=b is solvable --
well, actually,

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we had an answer in the
language of the column space.

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Can you remind me what that
answer is?

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That, that was like our answer
from earlier lecture.

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b had to be in the column
space.

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00:08:02 --> 00:08:08
Solvable if -- when -- exactly
when b is in the column space of

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A.

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Right?

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That just says that b has to be
a combination of the columns,

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and of course that's exactly
what the equation is looking

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for.

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So that -- now I want to answer
it -- the same answer but in

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different language.

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Another way to answer this --
if a combination of the rows of

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A gives the zero row,
and this was an example where

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it happened, some combination of
the rows of A produced the zero

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row -- then what's the
requirement on b?

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Since we're going to do the
same thing to both sides of all

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equations --
the same combination of the

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components of b has to give 0.

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Right?

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00:09:15 --> 00:09:21
That's -- so if there's a
combination of the rows that

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gives the zero row,
then the same combination of

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the entries of b must give 0.

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And this isn't the zero row,
that's the zero number.

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OK.

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Tthis is another way of saying
-- and it is not immediate,

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right, that these two
statements are equivalent.

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But somehow they must be,
because they're both equivalent

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to the solvability of the
system.

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OK.

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00:09:59 --> 00:10:03
So we've got this,
this sort of -- like question

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00:10:03 --> 00:10:07
zero is, does the system have a
solution?

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OK, I'll come back to discuss
that further.

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Let's go forward when it does.

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00:10:13 --> 00:10:15
When there is a solution.

147
00:10:15 --> 00:10:18
And so what's our job now?

148
00:10:18 --> 00:10:23
Abstractly we sit back and we
say, OK, there's a solution,

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finished.

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It exists.

151
00:10:25 --> 00:10:28
But we want to construct it.

152
00:10:28 --> 00:10:32
So what's the algorithm,
the sequence of steps to find

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the solution?

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That's what I -- and of course
the quiz and the final,

155
00:10:37 --> 00:10:42
I'm going to give you a system
Ax=b and I'm going to ask you

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00:10:42 --> 00:10:45
for the solution,
if there is one.

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00:10:45 --> 00:10:53
And so this algorithm that you
want to follow.

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OK, let's see.

159
00:10:55 --> 00:11:05
So what's the -- so now to find
the complete solution to Ax=b.

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00:11:05 --> 00:11:05
OK.

161
00:11:05 --> 00:11:14
Let me start by finding one
solution, one particular

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solution.

163
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I'm expecting that I can,
because my system of equations

164
00:11:22 --> 00:11:28
now, that last equation is zero
equals zero, so that's all fine.

165
00:11:28 --> 00:11:34.89
I really have two equations --
actually I've got four unknowns,

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00:11:34.89 --> 00:11:40
so I'm expecting to find not
only a solution but a whole

167
00:11:40 --> 00:11:41
bunch of them.

168
00:11:41 --> 00:11:45
But let's just find one.

169
00:11:45 --> 00:11:48
So step one,
a particular solution,

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00:11:48 --> 00:11:49
x particular.

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00:11:49 --> 00:11:52
How do I find one particular
solution?

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00:11:52 --> 00:11:55.87
Well, let me tell you how I,
how I find it.

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00:11:55.87 --> 00:11:59
So this is -- since there are
lots of solutions,

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00:11:59 --> 00:12:05.23
you could have your own way to
find a particular one.

175
00:12:05.23 --> 00:12:09
But this is a pretty natural
way.

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00:12:09 --> 00:12:14
Set all free variables to zero.

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00:12:14 --> 00:12:23
Since those free variables are
the guys that can be anything,

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the most convenient choice is
zero.

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00:12:28 --> 00:12:34
And then solve Ax=b for the
pivot variables.

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00:12:34 --> 00:12:41
So what does that mean in this
example?

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00:12:41 --> 00:12:44.02
Which are the free variables?

182
00:12:44.02 --> 00:12:48
Which, which are the variables
that we can assign freely and

183
00:12:48 --> 00:12:54.02
then there's one and only one
way to find the pivot variables?

184
00:12:54.02 --> 00:12:58
They're x2 and -- so x2 is
zero, because that's in a column

185
00:12:58 --> 00:13:03.52
without a pivot,
the second column has no pivot.

186
00:13:03.52 --> 00:13:06
And the -- what's the other
one?

187
00:13:06 --> 00:13:08
The fourth, x4 is zero.

188
00:13:08 --> 00:13:11
Because that,
those are the,

189
00:13:11 --> 00:13:12
the free ones.

190
00:13:12 --> 00:13:16
Those are in the columns with
no pivots.

191
00:13:16 --> 00:13:21
So you see what my -- so when I
knock -- when x2 and x4 are

192
00:13:21 --> 00:13:27.57
zero, I'm left with the -- what
I left with here?

193
00:13:27.57 --> 00:13:32
I'm just left with -- see,
now I'm not using the two free

194
00:13:32 --> 00:13:32
columns.

195
00:13:32 --> 00:13:35
I'm only using the pivot
columns.

196
00:13:35 --> 00:13:40.35
So I'm really left with x1 --
the first equation is just x1

197
00:13:40.35 --> 00:13:43
and two x3s should be the
right-hand side,

198
00:13:43 --> 00:13:46
which we picked to be a one.

199
00:13:46 --> 00:13:50
And the second equation is two
x3s, as it happened,

200
00:13:50 --> 00:13:53
turned out to be,
three.

201
00:13:53 --> 00:13:59
I just write it again here with
the x2 and the x4 knocked out,

202
00:13:59 --> 00:14:02.84
since we're set them to zero.

203
00:14:02.84 --> 00:14:09
And you see that we're back in
the normal case of having back

204
00:14:09 --> 00:14:13
-- where back substitution will
do it.

205
00:14:13 --> 00:14:18
So x3 is three halves,
and then we go back up and x1

206
00:14:18 --> 00:14:21
is one minus two x3.

207
00:14:21 --> 00:14:25
That's probably minus two.

208
00:14:25 --> 00:14:26
Good.

209
00:14:26 --> 00:14:33
So now we have the solution,
x particular is the vector

210
00:14:33 --> 00:14:38
minus two zero three halves
zero.

211
00:14:38 --> 00:14:39
OK, good.

212
00:14:39 --> 00:14:48
That's one particular solution,
and we should and could plug it

213
00:14:48 --> 00:14:53
into the original system.

214
00:14:53 --> 00:14:58
Really if -- on the quiz,
please, it's a good thing to

215
00:14:58 --> 00:14:58.68
do.

216
00:14:58.68 --> 00:15:02
So we did all this,
these, row operations,

217
00:15:02 --> 00:15:07
but this is supposed to solve
the original system,

218
00:15:07 --> 00:15:09
and I think it does.

219
00:15:09 --> 00:15:09
OK.

220
00:15:09 --> 00:15:13
So that's x particular which
we've got.

221
00:15:13 --> 00:15:16
So that's like what's new
today.

222
00:15:16 --> 00:15:20
The particular solution comes
-- first you check that you have

223
00:15:20 --> 00:15:23
zero equals zero,
so you're OK on the last

224
00:15:23 --> 00:15:23
equations.

225
00:15:23 --> 00:15:26
And then you set the free
variables to zero,

226
00:15:26 --> 00:15:30.76
solve for the pivot variables,
and you've got a particular

227
00:15:30.76 --> 00:15:34
solution, the particular
solution that has zero free

228
00:15:34 --> 00:15:35
variables.

229
00:15:35 --> 00:15:36
OK.

230
00:15:36 --> 00:15:43
Now -- but that's only one
solution, and now I'm looking

231
00:15:43 --> 00:15:44
for all.

232
00:15:44 --> 00:15:47.75
So how do I find the rest?

233
00:15:47.75 --> 00:15:55
The point is I can add on x --
anything out of the null space.

234
00:15:55 --> 00:16:03
We know how to find the vectors
in the null space --

235
00:16:03 --> 00:16:08
because we did it last time,
but I'll remind you what we

236
00:16:08 --> 00:16:08
got.

237
00:16:08 --> 00:16:10
And then I'll add.

238
00:16:10 --> 00:16:15
So the final result will be
that the complete solution --

239
00:16:15 --> 00:16:20
this is now the complete guy --
the complete solution is this

240
00:16:20 --> 00:16:24.7
one particular solution plus
any, any vector,

241
00:16:24.7 --> 00:16:29
all different vectors out of
the null space.

242
00:16:29 --> 00:16:30
xn, OK.

243
00:16:30 --> 00:16:36
Well why, why this pattern,
because this pattern shows up

244
00:16:36 --> 00:16:43
through all of mathematics,
because it shows up everywhere

245
00:16:43 --> 00:16:45
we have linear equations.

246
00:16:45 --> 00:16:50
Let me just put here the,
the reason.

247
00:16:50 --> 00:16:58
A xp, so that's x particular,
so what does Ax particular

248
00:16:58 --> 00:16:59
give?

249
00:16:59 --> 00:17:04.84
That gives the correct
right-hand side b.

250
00:17:04.84 --> 00:17:11
And what does A times an x in
the null space give?

251
00:17:11 --> 00:17:12
Zero.

252
00:17:12 --> 00:17:17.09
So I add, and I put in
parentheses.

253
00:17:17.09 --> 00:17:24
So xp plus xn is b plus zero,
which is b.

254
00:17:24 --> 00:17:26
So -- oh, what I saying?

255
00:17:26 --> 00:17:29
Let me just say it in words.

256
00:17:29 --> 00:17:33
If I have one solution,
I can add on anything in the

257
00:17:33 --> 00:17:38
null space, because anything in
the null space has a zero

258
00:17:38 --> 00:17:42
right-hand side,
and I still have the correct

259
00:17:42 --> 00:17:44
right-hand side B.

260
00:17:44 --> 00:17:47
So that's my system.

261
00:17:47 --> 00:17:50
That's my complete solution.

262
00:17:50 --> 00:17:56
Now let me write out what that
will be for this example.

263
00:17:56 --> 00:18:00
So in this example,
x general, x complete,

264
00:18:00 --> 00:18:04.6
the complete solution,
is x particular,

265
00:18:04.6 --> 00:18:10
which is minus two zero three
halves zero, with those zeroes

266
00:18:10 --> 00:18:15
in the free variable,
plus --

267
00:18:15 --> 00:18:19
you remember there were the
special solutions in the null

268
00:18:19 --> 00:18:24
space that had a one in the free
variables -- or one and zero in

269
00:18:24 --> 00:18:28.07
the free variables,
and then we filled in to find

270
00:18:28.07 --> 00:18:28
the others?

271
00:18:28 --> 00:18:33
I've forgotten what they were,
but maybe it was that.

272
00:18:33 --> 00:18:37
That was a special solution,
and then there was another

273
00:18:37 --> 00:18:42
special solution that had that
free variable zero and this free

274
00:18:42 --> 00:18:46
variable equal one,
and I have to fill those in.

275
00:18:46 --> 00:18:49
Let's see, can I remember how
those fill in?

276
00:18:49 --> 00:18:53
Maybe this was a minus two and
this was a two,

277
00:18:53 --> 00:18:53
possibly?

278
00:18:53 --> 00:18:56
I think probably that's right.

279
00:18:56 --> 00:18:58
I'm not -- yeah.

280
00:18:58 --> 00:19:00
Does that look write to you?

281
00:19:00 --> 00:19:04
I would have to remember what
are my equations.

282
00:19:04 --> 00:19:08
Can I, rather than go way back
to that board,

283
00:19:08 --> 00:19:12
let me remember the first
equation was two x3 plus two x4

284
00:19:12 --> 00:19:16
equaling zero now,
because I'm looking for the

285
00:19:16 --> 00:19:19
guys in the null space.

286
00:19:19 --> 00:19:23
So I set x4 to be one and the
second equation,

287
00:19:23 --> 00:19:29
that I didn't copy again,
gave me minus two for this and

288
00:19:29 --> 00:19:32
then -- yeah,
so I think that's right.

289
00:19:32 --> 00:19:36
Two minus four and two gives
zero, check.

290
00:19:36 --> 00:19:37
OK.

291
00:19:37 --> 00:19:40
Those were the special
solutions.

292
00:19:40 --> 00:19:45
What do we do to get the
complete solution?

293
00:19:45 --> 00:19:49
How do I get the complete
solution now?

294
00:19:49 --> 00:19:54
I multiply this by anything,
c1, say, and I multiply this by

295
00:19:54 --> 00:19:58
anything -- I take any
combination.

296
00:19:58 --> 00:20:02
Remember that's how we
described the null space?

297
00:20:02 --> 00:20:08
The null space consists of all
combinations of --

298
00:20:08 --> 00:20:12
so this is xn -- all
combinations of the special

299
00:20:12 --> 00:20:13
solutions.

300
00:20:13 --> 00:20:18
There were two special
solutions because there were two

301
00:20:18 --> 00:20:19
free variables.

302
00:20:19 --> 00:20:23
And we want to make that count
-- carefully now.

303
00:20:23 --> 00:20:25
Just while I'm up here.

304
00:20:25 --> 00:20:30
So there's, that's what the --
that's the kind of answer I'm

305
00:20:30 --> 00:20:31
looking for.

306
00:20:31 --> 00:20:34
Is there a constant multiplying
this guy?

307
00:20:34 --> 00:20:38.5
Is there a free constant that
multiplies x particular?

308
00:20:38.5 --> 00:20:39
No way.

309
00:20:39 --> 00:20:41
Right?
x particular solves A xp=b.

310
00:20:41 --> 00:20:45
I'm not allowed to multiply
that by three.

311
00:20:45 --> 00:20:49
But Axn, I'm allowed to
multiply xn by three,

312
00:20:49 --> 00:20:54
or add to another xn,
because I keep getting zero on

313
00:20:54 --> 00:20:56
the right.

314
00:20:56 --> 00:20:56
OK.

315
00:20:56 --> 00:21:00
So, so again,
xp is one particular guy.

316
00:21:00 --> 00:21:03
xn is a whole subspace.

317
00:21:03 --> 00:21:04
Right?

318
00:21:04 --> 00:21:10
It's one guy plus,
plus anything from a subspace.

319
00:21:10 --> 00:21:12
Let me draw it.

320
00:21:12 --> 00:21:15.34
Let me try to -- oh.

321
00:21:15.34 --> 00:21:21
I want to draw,
I want to graph all this -- I

322
00:21:21 --> 00:21:26
want to, I want to plot all
solutions.

323
00:21:26 --> 00:21:28
Now x.

324
00:21:28 --> 00:21:30.52
So what dimension I in?

325
00:21:30.52 --> 00:21:33
This is a unfortunate point.

326
00:21:33 --> 00:21:36
How many components does x
have?

327
00:21:36 --> 00:21:37
Four.

328
00:21:37 --> 00:21:39
There are four unknowns.

329
00:21:39 --> 00:21:45
So I have to draw a four
dimensional picture on this MIT

330
00:21:45 --> 00:21:47
cheap blackboard.

331
00:21:47 --> 00:21:47
OK.

332
00:21:47 --> 00:21:53
So here we go.
x1 -- Einstein could do it,

333
00:21:53 --> 00:22:01
but, this, this is -- those are
four perpendicular axes in --

334
00:22:01 --> 00:22:05
representing four dimensional
space.

335
00:22:05 --> 00:22:05
OK.

336
00:22:05 --> 00:22:08
Where are my solutions?

337
00:22:08 --> 00:22:12
Do my solutions form a
subspace?

338
00:22:12 --> 00:22:18
Does the set of solutions to
Ax=b form a subspace?

339
00:22:18 --> 00:22:20
No way.

340
00:22:20 --> 00:22:24
What does it actually look
like, though?

341
00:22:24 --> 00:22:26
A subspace is in this picture.

342
00:22:26 --> 00:22:28
This part is a subspace,
right?

343
00:22:28 --> 00:22:32.12
That part is some,
like, two dimensional,

344
00:22:32.12 --> 00:22:35.83
because I've got two
parameters, so it's -- I'm

345
00:22:35.83 --> 00:22:40
thinking of this null space as a
two dimensional subspace inside

346
00:22:40 --> 00:22:42.2
R^4.

347
00:22:42.2 --> 00:22:45
Now I have to tell you and will
tell you next time,

348
00:22:45 --> 00:22:49
what does it mean to say a
subspace, what's the dimension

349
00:22:49 --> 00:22:49
of a subspace.

350
00:22:49 --> 00:22:52.05
But you see what it's going to
be.

351
00:22:52.05 --> 00:22:55
It's the number of free
independent constants that we

352
00:22:55 --> 00:22:56
can choose.

353
00:22:56 --> 00:22:59
So somehow there'll be a two
dimensional subspace,

354
00:22:59 --> 00:23:02
not a line, and not a three
dimensional plane,

355
00:23:02 --> 00:23:05
but only a two dimensional guy.

356
00:23:05 --> 00:23:11
But it's doesn't go through the
origin because it goes through

357
00:23:11 --> 00:23:12
this point.

358
00:23:12 --> 00:23:18
So there's x particular.
x particular is somewhere here.

359
00:23:18 --> 00:23:20
x particular.

360
00:23:20 --> 00:23:27
So it's somehow a subspace --
can I try to draw it that way?

361
00:23:27 --> 00:23:35
It's a two dimensional subspace
that goes through x particular

362
00:23:35 --> 00:23:40
and then onwards by -- so
there's x particular,

363
00:23:40 --> 00:23:44
and I added on xn,
and there's x.

364
00:23:44 --> 00:23:46.74
There's x=xp+xn.

365
00:23:46.74 --> 00:23:54
But the xn was anywhere in this
subspace, so that filled out a

366
00:23:54 --> 00:23:56
plane.

367
00:23:56 --> 00:24:00
It's a subspace -- it's not a
subspace, what I saying?

368
00:24:00 --> 00:24:04
It's like a flat thing,
it's like a subspace,

369
00:24:04 --> 00:24:08
but it's been shifted,
away from the origin.

370
00:24:08 --> 00:24:10.46
It doesn't contain zero.

371
00:24:10.46 --> 00:24:10
OK.

372
00:24:10 --> 00:24:12
Thanks.

373
00:24:12 --> 00:24:15
That's the picture,
and that's the algorithm.

374
00:24:15 --> 00:24:20.39
So the algorithm is just go
through elimination and,

375
00:24:20.39 --> 00:24:25
find the particular solution,
and then find those special

376
00:24:25 --> 00:24:26
solutions.

377
00:24:26 --> 00:24:27.55
You can do that.

378
00:24:27.55 --> 00:24:31
Let me take our time here in
the lecture to think,

379
00:24:31 --> 00:24:34
about the bigger picture.

380
00:24:34 --> 00:24:42
So let me think about -- so
this is my pattern.

381
00:24:42 --> 00:24:52.03
Now I want to think -- I want
to ask you about a question -- I

382
00:24:52.03 --> 00:24:56
want to ask you some questions.

383
00:24:56 --> 00:25:06
So when I mean think bigger,
I mean I'll think about an m by

384
00:25:06 --> 00:25:11
n matrix A of rank r.

385
00:25:11 --> 00:25:11
OK.

386
00:25:11 --> 00:25:14
What's our definition of rank?

387
00:25:14 --> 00:25:19
Our current definition of rank
is number of pivots.

388
00:25:19 --> 00:25:19
OK.

389
00:25:19 --> 00:25:23
First of all,
how are these numbers related?

390
00:25:23 --> 00:25:27
Can you tell me a relation
between r and m?

391
00:25:27 --> 00:25:32
If I have m rows in the matrix
and R pivots,

392
00:25:32 --> 00:25:35
--
then I certainly know,

393
00:25:35 --> 00:25:41
always -- what relation do I
know between r and m?

394
00:25:41 --> 00:25:44
r is less or equal,
right?

395
00:25:44 --> 00:25:49
Because I've got m rows,
I can't have more than m

396
00:25:49 --> 00:25:54
pivots, I might have m and I
might have fewer.

397
00:25:54 --> 00:25:57
Also, I've got n columns.

398
00:25:57 --> 00:26:02
So what's the relation between
r and n?

399
00:26:02 --> 00:26:04.81
It's the same,
less or equal,

400
00:26:04.81 --> 00:26:08
because a column can't have
more than one pivot.

401
00:26:08 --> 00:26:12
So I can't have more than n
pivots altogether.

402
00:26:12 --> 00:26:13
OK, OK.

403
00:26:13 --> 00:26:15
So I have an m by n matrix of
rank r.

404
00:26:15 --> 00:26:19
And I always know r less than
or equal to m,

405
00:26:19 --> 00:26:22
r less than or equal to n.

406
00:26:22 --> 00:26:28
Now I'm specially interested in
the case of full rank,

407
00:26:28 --> 00:26:32
when the rank r is as big as it
can be.

408
00:26:32 --> 00:26:38
Well, I guess I've got two
separate possibilities here,

409
00:26:38 --> 00:26:43
depending on what these numbers
m and n are.

410
00:26:43 --> 00:26:50
So let me talk about the case
of full column rank.

411
00:26:50 --> 00:26:53
And by that I mean r=n.

412
00:26:53 --> 00:27:00
And I want to ask you,
what does that imply about our

413
00:27:00 --> 00:27:02.16
solutions?

414
00:27:02.16 --> 00:27:08
What does that tell us about
the null space?

415
00:27:08 --> 00:27:16
What does that tell us about,
the complete solution?

416
00:27:16 --> 00:27:19
OK, so what does that mean?

417
00:27:19 --> 00:27:26
So I want to ask you,
well, OK, if the rank is n,

418
00:27:26 --> 00:27:28
what does that mean?

419
00:27:28 --> 00:27:34
That means there's a pivot in
every column.

420
00:27:34 --> 00:27:39
So how many pivot variables are
there?

421
00:27:39 --> 00:27:40
n.

422
00:27:40 --> 00:27:43.62
All the columns have pivots in
this case.

423
00:27:43.62 --> 00:27:46
So how many free variables are
there?

424
00:27:46 --> 00:27:47
None at all.

425
00:27:47 --> 00:27:51
So no free variables.
r=n, no free variables.

426
00:27:51 --> 00:27:57
So what does that tell us about
what's going to happen then in

427
00:27:57 --> 00:28:00
our, in our little algorithms?

428
00:28:00 --> 00:28:05
What will be in the null space?

429
00:28:05 --> 00:28:10
The null space of A has got
what in it?

430
00:28:10 --> 00:28:13
Only the zero vector.

431
00:28:13 --> 00:28:21
There are no free variables to
give other values to.

432
00:28:21 --> 00:28:28
So the null space is only the
zero vector.

433
00:28:28 --> 00:28:32
And what about our solution to
Ax=b?

434
00:28:32 --> 00:28:34
Solution to Ax=b?

435
00:28:34 --> 00:28:38
What, what's the story on that
one?

436
00:28:38 --> 00:28:43
So now that's coming from
today's lecture.

437
00:28:43 --> 00:28:49
The solution x is -- what's the
complete solution?

438
00:28:49 --> 00:28:53
It's just x particular,
right?

439
00:28:53 --> 00:29:00
If, if, if there is an x,
if there is a solution.

440
00:29:00 --> 00:29:03
It's x equal x particular.

441
00:29:03 --> 00:29:10
There's nothing -- you know,
there's just one solution.

442
00:29:10 --> 00:29:13
If there's one at all.

443
00:29:13 --> 00:29:20.86
So it's unique solution --
unique means only one -- unique

444
00:29:20.86 --> 00:29:26
solution if it exists,
if it exists.

445
00:29:26 --> 00:29:32
In other words,
I would say -- let me put it a

446
00:29:32 --> 00:29:33
different way.

447
00:29:33 --> 00:29:38
There're either zero or one
solutions.

448
00:29:38 --> 00:29:42
This is all in this case r=n.

449
00:29:42 --> 00:29:49
So I'm -- because many,
many applications in reality,

450
00:29:49 --> 00:29:57
the columns will be what I'll
later call independent.

451
00:29:57 --> 00:30:01
And we'll have,
nothing to look for in the null

452
00:30:01 --> 00:30:06
space, and we'll only have
particular solutions.

453
00:30:06 --> 00:30:07
OK.

454
00:30:07 --> 00:30:10
Everybody see that possibility?

455
00:30:10 --> 00:30:13
But I need an example,
right?

456
00:30:13 --> 00:30:16
So let me create an example.

457
00:30:16 --> 00:30:22
What sort of a matrix --
what's the shape of a matrix

458
00:30:22 --> 00:30:25
that has full column rank?

459
00:30:25 --> 00:30:29.47
So can I squeeze in an,
an example here?

460
00:30:29.47 --> 00:30:30
If it exists.

461
00:30:30 --> 00:30:36
Let me put in an example,
and it's just the right space

462
00:30:36 --> 00:30:38
to put in an example.

463
00:30:38 --> 00:30:44
Because the example will be
like tall and thin.

464
00:30:44 --> 00:30:50
It will have -- well,
I mean, here's an example,

465
00:30:50 --> 00:30:54
one two six five,
three one one one.

466
00:30:54 --> 00:30:56
Brilliant example.

467
00:30:56 --> 00:30:57
OK.

468
00:30:57 --> 00:31:02
So there's a matrix A,
and what's its rank?

469
00:31:02 --> 00:31:05
What's the rank of that matrix?

470
00:31:05 --> 00:31:13
How many pivots will I find if
I do elimination?

471
00:31:13 --> 00:31:14.07
Two, right?

472
00:31:14.07 --> 00:31:14.41
Two.

473
00:31:14.41 --> 00:31:19
I see a pivot there -- oh
certainly those two columns are

474
00:31:19 --> 00:31:22
headed off in different
directions.

475
00:31:22 --> 00:31:26
When I do elimination,
I'll certainly get another

476
00:31:26 --> 00:31:30
pivot here, fine,
and I can use those to clean

477
00:31:30 --> 00:31:32
out below and above.

478
00:31:32 --> 00:31:37
So the -- actually,
tell me what its row reduced

479
00:31:37 --> 00:31:40
row echelon form would be.

480
00:31:40 --> 00:31:46
Can you carry that,
that elimination process to the

481
00:31:46 --> 00:31:47
bitter end?

482
00:31:47 --> 00:31:50
So what do, what does that
mean?

483
00:31:50 --> 00:31:57
I subtract a multiple of this
row from these rows.

484
00:31:57 --> 00:31:59
So I clean up,
all zeros there.

485
00:31:59 --> 00:32:01
Then I've got some pivot here.

486
00:32:01 --> 00:32:03
What do I do with that?

487
00:32:03 --> 00:32:06
I go subtract it below and
above, and then I divide

488
00:32:06 --> 00:32:09
through, and what's R for that
example?

489
00:32:09 --> 00:32:13
Maybe I can -- you'll allow me
to put that just here in the

490
00:32:13 --> 00:32:15
next board.

491
00:32:15 --> 00:32:20
What's the row reduced echelon
form, just out of practice,

492
00:32:20 --> 00:32:22.33
for that matrix?

493
00:32:22.33 --> 00:32:25
It's got ones in the pivots.

494
00:32:25 --> 00:32:30
It's got the identity matrix,
a little two by two identity

495
00:32:30 --> 00:32:33
matrix, and below it all zeros.

496
00:32:33 --> 00:32:38
That's a matrix that really has
two independent rows,

497
00:32:38 --> 00:32:42
and they're the first two,
actually.

498
00:32:42 --> 00:32:45
The first two rows are
independent.

499
00:32:45 --> 00:32:48
They're not in the same
direction.

500
00:32:48 --> 00:32:52
But the other rows are
combinations of the first two,

501
00:32:52 --> 00:32:55
so -- is there always a
solution to Ax=b?

502
00:32:55 --> 00:32:58.37
Tell me what's the picture
here?

503
00:32:58.37 --> 00:33:02
For this matrix A,
this is a case of full column

504
00:33:02 --> 00:33:03
rank.

505
00:33:03 --> 00:33:07
The two columns are -- give two
pivots.

506
00:33:07 --> 00:33:09
There's nothing in the null
space.

507
00:33:09 --> 00:33:15
There's no combination of those
columns that gives the zero

508
00:33:15 --> 00:33:18
column except the zero zero
combination.

509
00:33:18 --> 00:33:21
So there's nothing in the null
space.

510
00:33:21 --> 00:33:26
But is there always a solution
to A X equal B?

511
00:33:26 --> 00:33:29.34
What's up with A X equal B?

512
00:33:29.34 --> 00:33:32
I've got four,
four equations here,

513
00:33:32 --> 00:33:34
and only two Xs.

514
00:33:34 --> 00:33:37
So the answer is certainly no.

515
00:33:37 --> 00:33:40
There's not always a solution.

516
00:33:40 --> 00:33:45
I may have zero solutions,
and if I make a random choice,

517
00:33:45 --> 00:33:49
I'll have zero solutions.

518
00:33:49 --> 00:33:53
Or if I make a great particular
choice of the right-hand side,

519
00:33:53 --> 00:33:58
which just happens to be a
combination of those two guys --

520
00:33:58 --> 00:34:02
like tell me one right-hand side
that would have a solution.

521
00:34:02 --> 00:34:07.04
Tell me a right-hand side that
would have a solution.

522
00:34:07.04 --> 00:34:09.15
Well, 0 0 0 0,
OK.

523
00:34:09.15 --> 00:34:11
No prize for that one.

524
00:34:11 --> 00:34:12
Tell me another one.

525
00:34:12 --> 00:34:17
Another right-hand side that
has a solution would be 4 3 7 6.

526
00:34:17 --> 00:34:20.13
I could add the two columns.

527
00:34:20.13 --> 00:34:20
Right?

528
00:34:20 --> 00:34:24
What would be the total
complete solution if the

529
00:34:24 --> 00:34:27
right-hand side was 4 3 7 6?

530
00:34:27 --> 00:34:31
There would be the particular
solution one one,

531
00:34:31 --> 00:34:36
one of that column plus one of
that, and that would be the only

532
00:34:36 --> 00:34:36
solution.

533
00:34:36 --> 00:34:40
So there would be -- x
particular would be one one in

534
00:34:40 --> 00:34:45
the case when the right side is
the sum of those two columns,

535
00:34:45 --> 00:34:47
and that's it.

536
00:34:47 --> 00:34:52
So that would be a case with
one solution.

537
00:34:52 --> 00:34:53
OK.

538
00:34:53 --> 00:35:00
That, this is the typical setup
with full column rank.

539
00:35:00 --> 00:35:03
Now I go to full row rank.

540
00:35:03 --> 00:35:11
You see the sort of natural
symmetry of this discussion.

541
00:35:11 --> 00:35:14
Full row rank means r=m.

542
00:35:14 --> 00:35:21
So this is what I'm interested
in now, r=m.

543
00:35:21 --> 00:35:25
OK, what's up with that?

544
00:35:25 --> 00:35:27
How many pivots?
m.

545
00:35:27 --> 00:35:35
So what happens when we do
elimination in that case?

546
00:35:35 --> 00:35:39
I'm going to get m pivots.

547
00:35:39 --> 00:35:43
So every row has a pivot,
right?

548
00:35:43 --> 00:35:46
Every row has a pivot.

549
00:35:46 --> 00:35:50
Then what about solvability?

550
00:35:50 --> 00:36:00
What about this business of --
for which right-hand sides can

551
00:36:00 --> 00:36:02.28
I solve it?

552
00:36:02.28 --> 00:36:05.2
So that's my question.

553
00:36:05.2 --> 00:36:10
I can solve Ax=b for which
right-hand sides?

554
00:36:10 --> 00:36:14
Do you see what's coming?

555
00:36:14 --> 00:36:19
I do elimination,
I don't get any zero rows.

556
00:36:19 --> 00:36:26
So there aren't any
requirements on b.

557
00:36:26 --> 00:36:30
I can solve Ax=b for every b.

558
00:36:30 --> 00:36:36
I can solve Ax=b for every
right-hand side.

559
00:36:36 --> 00:36:42
So this is the existence,
exists a solution.

560
00:36:42 --> 00:36:49
Now tell me,
so the, u- u- so every row has

561
00:36:49 --> 00:36:51
a pivot in it.

562
00:36:51 --> 00:36:57
So how many free variables are
there?

563
00:36:57 --> 00:37:04
How many free variables in this
case?

564
00:37:04 --> 00:37:15
If I had n variables to start
with, how many are used up by

565
00:37:15 --> 00:37:20.61
pivot variables?
r, which is m.

566
00:37:20.61 --> 00:37:29
So I'm left with,
left with n-r free variables.

567
00:37:29 --> 00:37:31
OK.

568
00:37:31 --> 00:37:35
So this case of full row rank I
can always solve,

569
00:37:35 --> 00:37:40
and then this tells me how many
variables are free,

570
00:37:40 --> 00:37:43.37
and this is of course n-m.

571
00:37:43.37 --> 00:37:46
This is n-m free variables.

572
00:37:46 --> 00:37:47
Can I do an example?

573
00:37:47 --> 00:37:53.46
You know, the best way for me
to do an example is just to

574
00:37:53.46 --> 00:37:56
transpose that example.

575
00:37:56 --> 00:38:00
So let me take,
let me take that matrix that

576
00:38:00 --> 00:38:05
had column one two six five and
make it a row.

577
00:38:05 --> 00:38:10
And let me take three one one
one as the second row.

578
00:38:10 --> 00:38:14.06
And let me ask you,
this is my matrix A,

579
00:38:14.06 --> 00:38:15
what's its rank?

580
00:38:15 --> 00:38:19
What's the rank of that matrix?

581
00:38:19 --> 00:38:22.4
Sorry to ask,
but not sorry really,

582
00:38:22.4 --> 00:38:25
because we're just getting the
idea of rank.

583
00:38:25 --> 00:38:28
What's the rank of that matrix?

584
00:38:28 --> 00:38:29
Two, exactly,
two.

585
00:38:29 --> 00:38:31
There will be two pivots.

586
00:38:31 --> 00:38:35
What will the row reduced
echelon form be?

587
00:38:35 --> 00:38:37
Anybody know that one?

588
00:38:37 --> 00:38:41
Actually, tell me not only --
you have to tell me not only

589
00:38:41 --> 00:38:45
the, there'll be two pivots but
which will be the pivot columns.

590
00:38:45 --> 00:38:48
Which columns of this matrix
will be pivot columns?

591
00:38:48 --> 00:38:52
So the first column is fine,
and then I go on to the next

592
00:38:52 --> 00:38:53.84
column, and what do I get?

593
00:38:53.84 --> 00:38:56
Do I get a second pivot out of
--

594
00:38:56 --> 00:39:01
will I get a second pivot in
this position?

595
00:39:01 --> 00:39:01
Yes.

596
00:39:01 --> 00:39:06
So the pivots,
when I get all the way to R,

597
00:39:06 --> 00:39:07.65
will be there.

598
00:39:07.65 --> 00:39:10
And here will be some numbers.

599
00:39:10 --> 00:39:15.59
This is the part that I
previously called F.

600
00:39:15.59 --> 00:39:22.4
This is the part that --
the pivot columns in R will be

601
00:39:22.4 --> 00:39:24
the identity matrix.

602
00:39:24 --> 00:39:28
There are no zero rows,
no zero rows,

603
00:39:28 --> 00:39:30
because the rank is two.

604
00:39:30 --> 00:39:33
But there'll be stuff over
here.

605
00:39:33 --> 00:39:38
And that will,
enter the special solutions and

606
00:39:38 --> 00:39:41
the null space.

607
00:39:41 --> 00:39:41
OK.

608
00:39:41 --> 00:39:48
So this is a typical matrix
with r=m smaller than n.

609
00:39:48 --> 00:39:53
Now finally I've got a space
here for r=m=n.

610
00:39:53 --> 00:40:01
I'm off in the corner here with
the most important case of all.

611
00:40:01 --> 00:40:05
So what's up with this matrix?

612
00:40:05 --> 00:40:08
So let me give an example.

613
00:40:08 --> 00:40:14
OK, brilliant example,
1 2 3 1.

614
00:40:14 --> 00:40:20
Tell me what -- how do I
describe a matrix that has rank

615
00:40:20 --> 00:40:21
r=m=n?

616
00:40:21 --> 00:40:26
So the matrix is square,
right, it's a square matrix.

617
00:40:26 --> 00:40:31.83
And if I know its rank is --
it's full rank,

618
00:40:31.83 --> 00:40:32
now.

619
00:40:32 --> 00:40:39.69
I don't have to say full column
rank or full row rank --

620
00:40:39.69 --> 00:40:43
I just say full rank,
because the count,

621
00:40:43 --> 00:40:47
column count and the row count
are the same,

622
00:40:47 --> 00:40:50
and the rank is as big as it
can be.

623
00:40:50 --> 00:40:54
And what kind of a matrix have
I got?

624
00:40:54 --> 00:40:55.57
It's invertible.

625
00:40:55.57 --> 00:40:59
So that's exactly the
invertible matrices.

626
00:40:59 --> 00:41:05
r=m=n means the --
what's the row echelon form,

627
00:41:05 --> 00:41:10
the reduced row echelon form,
for an invertible matrix?

628
00:41:10 --> 00:41:13
For a square,
nice, square,

629
00:41:13 --> 00:41:15
invertible matrix?

630
00:41:15 --> 00:41:16
It's I.

631
00:41:16 --> 00:41:16
Right.

632
00:41:16 --> 00:41:22
So you see that the,
the good matrices are the ones

633
00:41:22 --> 00:41:27.36
that kind of come out trivially
in R.

634
00:41:27.36 --> 00:41:31.66
You reduce them all the way to
the identity matrix.

635
00:41:31.66 --> 00:41:35
What's the null space for this,
for this matrix?

636
00:41:35 --> 00:41:38
Can I just hammer away with
questions?

637
00:41:38 --> 00:41:42
What's the null space for this
matrix?

638
00:41:42 --> 00:41:47
The null space of that matrix
is the zero vector only.

639
00:41:47 --> 00:41:50
The zero vector only.

640
00:41:50 --> 00:41:55.03
What are the conditions to
solve Ax=b?

641
00:41:55.03 --> 00:41:59
Which right-hand sides b are
OK?

642
00:41:59 --> 00:42:04
If I want to solve Ax=b for
this example,

643
00:42:04 --> 00:42:09.08
so A is this,
b is b1 b2, what are the

644
00:42:09.08 --> 00:42:13
conditions on b1 and b2?

645
00:42:13 --> 00:42:15
None at all,
right.

646
00:42:15 --> 00:42:19
So this is the case,
this is the case where I can

647
00:42:19 --> 00:42:24
solve -- so I've coming back
here, I can -- since the rank

648
00:42:24 --> 00:42:27
equals m, I can solve for every
b.

649
00:42:27 --> 00:42:32
And since the rank is also n,
there's a unique solution.

650
00:42:32 --> 00:42:37
Let me summarize the whole
picture here.

651
00:42:37 --> 00:42:39
Here's the whole picture.

652
00:42:39 --> 00:42:42
I could have r=m=n.

653
00:42:42 --> 00:42:47.86
This is the case where this is
the identity matrix.

654
00:42:47.86 --> 00:42:53
And this is the case where
there is one solution.

655
00:42:53 --> 00:42:58.59
That's the square invertible
chapter two case.

656
00:42:58.59 --> 00:43:01
Now we're into chapter three.

657
00:43:01 --> 00:43:06
We could have r=m smaller than
n.

658
00:43:06 --> 00:43:14
Now that's what we had over
there, and the row echelon form

659
00:43:14 --> 00:43:20
looked like the identity with
some zero rows.

660
00:43:20 --> 00:43:28.68
And that was the case where
there are zero or one solution.

661
00:43:28.68 --> 00:43:33
When I say solution I mean to
Ax=b.

662
00:43:33 --> 00:43:39
So this case,
there's always one.

663
00:43:39 --> 00:43:43
This case there's zero or one.

664
00:43:43 --> 00:43:50
And now let me take the case of
full column rank,

665
00:43:50 --> 00:43:53
but some, extra rows.

666
00:43:53 --> 00:44:00.95
So now R has -- well,
the identity -- I'm almost

667
00:44:00.95 --> 00:44:08
tempted to write the identity
matrix and then F,

668
00:44:08 --> 00:44:12
but that isn't necessarily
right.

669
00:44:12 --> 00:44:18
I have -- is that right?

670
00:44:18 --> 00:44:20
Am I getting this correct here?

671
00:44:20 --> 00:44:21
Oh, I'm not!

672
00:44:21 --> 00:44:22
My God!

673
00:44:22 --> 00:44:25
This is the case R equals n,
the columns,

674
00:44:25 --> 00:44:27
the columns are,
are OK.

675
00:44:27 --> 00:44:31
That's the case that was on
that board, r=n,

676
00:44:31 --> 00:44:33
full column rank.

677
00:44:33 --> 00:44:37
Now I want the case where m is
smaller than n and I've got

678
00:44:37 --> 00:44:40
extra columns.

679
00:44:40 --> 00:44:40
OK.

680
00:44:40 --> 00:44:43
There we go.

681
00:44:43 --> 00:44:54
So this is now the case of full
row rank, and it looks like I F

682
00:44:54 --> 00:45:06
except that I can't be sure that
the pivot columns are the first

683
00:45:06 --> 00:45:09
columns.

684
00:45:09 --> 00:45:14
So the I and the F,
could be partly mixed into the

685
00:45:14 --> 00:45:14
I.

686
00:45:14 --> 00:45:17
Can I write that with just like
that?

687
00:45:17 --> 00:45:22
So the F could be sort of
partly into the I if the first

688
00:45:22 --> 00:45:26
columns weren't the pivot
columns.

689
00:45:26 --> 00:45:30.53
Now how many solutions in this
case?

690
00:45:30.53 --> 00:45:32
There's always a solution.

691
00:45:32 --> 00:45:34
This is the existence case.

692
00:45:34 --> 00:45:37.18
There's always a solution.

693
00:45:37.18 --> 00:45:39
We're not getting any zero
rows.

694
00:45:39 --> 00:45:42
There are no zero rows here.

695
00:45:42 --> 00:45:46
So there's always either one or
infinitely many solutions.

696
00:45:46 --> 00:45:47
OK.

697
00:45:47 --> 00:45:51.49
Actually, I guess there's
always an infinite number,

698
00:45:51.49 --> 00:45:56
because we always have some
null space to deal with.

699
00:45:56 --> 00:46:04
Then the final case is where r
is smaller than m and smaller

700
00:46:04 --> 00:46:05
than n.

701
00:46:05 --> 00:46:05
OK.

702
00:46:05 --> 00:46:13
Now that's the case where R is
the identity with some free

703
00:46:13 --> 00:46:17
stuff but with some zero rows
too.

704
00:46:17 --> 00:46:25
And that's the case where
there's either no solution --

705
00:46:25 --> 00:46:31
because we didn't get a zero
equals zero for some bs,

706
00:46:31 --> 00:46:34
or infinitely many solutions.

707
00:46:34 --> 00:46:35
OK.

708
00:46:35 --> 00:46:41
Do you -- this board really
summarizes the lecture,

709
00:46:41 --> 00:46:45
and this sentence summarizes
the lecture.

710
00:46:45 --> 00:46:53
The rank tells you everything
about the number of solutions.

711
00:46:53 --> 00:46:58
That number,
the rank r, tells you all the

712
00:46:58 --> 00:47:04
information except the exact
entries in the solutions.

713
00:47:04 --> 00:47:08
For that you go to the matrix.

714
00:47:08 --> 00:47:09
OK, good.

715
00:47:09 --> 00:47:12
Have a great weekend,
and I'll see you on Monday.