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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?
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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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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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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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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.
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OK.
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So we do elimination on that.
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Can we just do elimination
quickly?
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The first pivot is fine,
I subtract two of this away
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from this, three of this away
from this, so I have 1 2 2 2 b1.
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Two of those away will give me
0 0 2 and 4, and that was b2
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minus two b1.
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I, I have to do the same thing
to that third,
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that last column.
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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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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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And let me get the final result
-- well, let me -- can I do it
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by eraser?
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We're capable of subtracting
this row from this row,
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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?
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I think I have b3 minus a b2,
and I had minus three b1s.
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This is going to,
it's going to be a minus a b1.
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00:05:25 --> 00:05:29
Oh yeah that's exactly what I
expect.
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So now the -- what's the last
equation?
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The last equation,
this represented by this zero
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row, that last equation is,
says 0 equals b3 minus b2 minus
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b1.
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So that's the condition for
solvability.
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That's the condition on the
right-hand side that we
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expected.
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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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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?
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The b1 will still be a 1.
b2 would be 5 minus 2,
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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.
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OK.
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So the last equation is OK now.
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And I can proceed to solve the
two equations that are really
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there with four unknowns.
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OK, I, I, I want to do that,
so this, this b is OK.
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It allows a solution.
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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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solvable.
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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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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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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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So we've got this,
this sort of -- like question
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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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When there is a solution.
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And so what's our job now?
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Abstractly we sit back and we
say, OK, there's a solution,
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finished.
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It exists.
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But we want to construct it.
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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,
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I'm going to give you a system
Ax=b and I'm going to ask you
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for the solution,
if there is one.
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And so this algorithm that you
want to follow.
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OK, let's see.
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00:10:55 --> 00:11:05
So what's the -- so now to find
the complete solution to Ax=b.
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OK.
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Let me start by finding one
solution, one particular
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solution.
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I'm expecting that I can,
because my system of equations
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now, that last equation is zero
equals zero, so that's all fine.
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I really have two equations --
actually I've got four unknowns,
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so I'm expecting to find not
only a solution but a whole
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bunch of them.
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But let's just find one.
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00:11:45 --> 00:11:48
So step one,
a particular solution,
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x particular.
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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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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.
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But this is a pretty natural
way.
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Set all free variables to zero.
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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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And then solve Ax=b for the
pivot variables.
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So what does that mean in this
example?
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Which are the free variables?
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Which, which are the variables
that we can assign freely and
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00:12:48 --> 00:12:54.02
then there's one and only one
way to find the pivot variables?
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00:12:54.02 --> 00:12:58
They're x2 and -- so x2 is
zero, because that's in a column
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00:12:58 --> 00:13:03.52
without a pivot,
the second column has no pivot.
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00:13:03.52 --> 00:13:06
And the -- what's the other
one?
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00:13:06 --> 00:13:08
The fourth, x4 is zero.
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00:13:08 --> 00:13:11
Because that,
those are the,
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00:13:11 --> 00:13:12
the free ones.
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00:13:12 --> 00:13:16
Those are in the columns with
no pivots.
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00:13:16 --> 00:13:21
So you see what my -- so when I
knock -- when x2 and x4 are
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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.
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00:13:35 --> 00:13:40.35
So I'm really left with x1 --
the first equation is just x1
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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.
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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.