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OK, here's linear
algebra lecture seven.
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I've been talking
about vector spaces
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and specially the
null space of a matrix
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and the column
space of a matrix.
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What's in those spaces.
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Now I want to actually
describe them.
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How do you describe
all the vectors
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that are in those spaces?
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How do you compute these things?
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So this is the, turning
the idea, the definition,
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into an algorithm.
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What's the algorithm
for solving A x =0?
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So that's the null space
that I'm interested in.
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So can I take a particular
matrix A and describe
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the natural algorithm, and I'll
execute it for that matrix --
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here we go.
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So let me take the
matrix as an example.
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So we're definitely talking
rectangular matrices in this
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chapter.
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So I'll make, I'll
have four columns.
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And three rows.
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Two four six eight and
three six eight ten.
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OK.
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If I just look at those
columns, and rows, well,
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I notice right away
that column two
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is a multiple of column one.
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It's in the same
direction as column one.
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It's not independent.
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I'll expect to discover
that in the process.
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Actually, with rows, I notice
that that row plus this row
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gives the third row.
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So the third row
is not independent.
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So, all that should
come out of elimination.
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So now what I --
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my algorithm is elimination,
but extended now
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to the rectangular
case, where we
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have to continue even if there's
zeros in the pivot position,
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we go on.
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OK, so let me execute
elimination for that matrix.
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My goal is always, while
I'm doing elimination --
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I'm not changing the null space.
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That's very important, right?
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I'm solving A x equals
zero by elimination,
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and when I do these operations
that you already know,
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when I subtract
a multiple of one
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equation from another equation,
I'm not changing the solutions.
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So I'm not changing
the null space.
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Actually, I changing the
column space, as you'll see.
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So you have to pay attention.
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What does elimination
leave unchanged?
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And the answer is the solutions
to the system are not changed
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because I'm doing
the same thing to --
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I'm doing a legitimate
operations on the equations.
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Of course, on the right
hand side it's always zero,
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and I don't plan to
write zero all the time.
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OK, so I'm really just
working on the left side,
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but the right side is, is
keeping up always zeros.
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OK, so what's elimination?
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Well, you know where
the first pivot is
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and you know what to do
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with it.
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So can I just take the
first step below here?
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So that pivot row is fine.
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I take two times that row away
from this one and I get zero
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zero.
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That's signaling a difficulty.
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Two, two twos away from the
six leaves me with a two.
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Two twos away from the
eight leaves me with a four.
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And now three of
those away from here
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is zero, again another
zero, three twos away
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from that eight is the
two, three twos away
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from that ten is a four.
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OK.
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That's the first
stage of elimination.
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I've got the first
column straight.
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So of course I move on
to the second column.
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I look in that
position, I see a zero.
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I look below it,
hoping for a non-zero
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that I can do a row exchange.
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But it's zero below.
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So that's telling me that
that column is -- well,
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what it's really going to be
telling me is that that column
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is a combination of
the earlier columns.
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It's that second column is
dependent on the earlier
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columns.
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But I don't stop to think here.
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In that column
there's nothing to do.
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I go on to the next.
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So here's the next pivot.
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So there's the first pivot
and there's the second pivot,
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and I just keep this
elimination going downwards.
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So, so the next step
keeps the first row,
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keeps the second
row with its pivot,
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so I've got my two pivots,
and does elimination
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to clear out the column
below that pivot.
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So actually you see
the multiplier is one.
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It subtracts row
two from row three
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and produces a row of zeros.
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OK.
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That I would call
that matrix U, right?
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That's our upper --
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well, I can't quite
say upper triangular.
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Maybe upper -- I don't
know -- upper something.
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It's in this so-called
echelon form.
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The word echelon means,
like, staircase form.
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It's the, the non-zeros
come in that staircase form.
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If there was another pivot
here, then the staircase
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would include that.
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But here's a case where
we have two pivots only.
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OK, so actually we've already
discovered the most important
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number about this matrix.
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The number of pivots is two.
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That number we will call
the rank of the matrix.
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So let me put immediately.
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The rank of A --
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so I'm telling
you what this word
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rank means in the algorithm.
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It's equal to the
number of pivots.
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And in this case, two.
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OK, for me that number
two is the crucial number.
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OK, now I go to -- you remember
I'm always solving A x equals
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zero, but now I can solve
U x equals zero, right?
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Same solution, same null space.
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OK.
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So I could stop here --
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why don't I stop here and
do the back substitution.
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So now I have to ask you, how
do I describe the solutions?
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There are solutions,
right, to A x equals zero.
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I knew there would be.
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I had three equations
in four unknowns.
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I certainly expected
some solutions.
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Now I want to see what are they.
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OK, here's the critical step.
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I refer to it up here
as separating out
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the pivot variables, the pivot
columns, which are these two.
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Here I have two pivot columns.
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Those, obviously, they're
the columns with the pivots.
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So I have two pivot columns.
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And I have the other
columns, I'll call free.
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These are free columns, OK.
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Why do I use those words?
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Why do I use that word free?
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Because now I want to write, I
want to find the solutions to U
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x equals zero.
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Here is the way I do it.
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These free columns I can assign
any number freely to those --
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to the variables x2 and
x4, the ones that multiply
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columns two and four.
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So I can assign anything
I like to x2 and x4.
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And then I can solve the
equations for x1 and x3.
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Let me say that again.
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If I give -- let
me, let me assign.
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So, so one particular x is
to assign, say, the value one
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to the, to x2, and zero to x4.
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Those are -- that
was a free choice,
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but it's a convenient choice.
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OK.
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Now I want to solve
U x equals zero
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and find numbers one and
three, complete the solution.
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Can I write down --
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let's see.
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OK.
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Shall we just remember what
U x equals zero represents?
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What are my equations?
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That first equation
is x1 plus just --
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I'm just saying what are
these matrices meaning.
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That's the first equation.
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And the second equation
was 2x3 + 4x4=0.
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Those are my two equations.
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OK.
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Now the point is I can find x1
and x3 by back substitution.
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So we're building on
what we already know.
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The new thing is that there
were some free variables that I
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could give any numbers to.
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And I'm systematically going
to make a choice like this, Now
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what is x3?
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1 and 0.
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Let's, let's go
backwards, back up.
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I look at the last equation.
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x3 is zero, from
the last equation,
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because, because x4
we've set x4 to zero,
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and then we get x3 as zero.
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OK.
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Now we set x2 to be
one, so what is x1?
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Negative two, right?
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So then I have negative two
plus two, zero and zero,
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correctly giving zero.
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There is a vector
in the null space.
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There is a solution
to A x equals zero.
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In fact, what solution is that?
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That simply says that minus
two times the first column
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plus one times the second
column is the zero column.
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Of course that's right.
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Minus two of that column plus
one of that, or minus two
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of that plus one of that.
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That solution is -- that, that's
just what we saw immediately,
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that the second column is twice
as big as the first column.
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OK, tell me some more
vectors in the null space.
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I found one.
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Tell me, how to get a bunch more
immediately out of that one.
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Just take multiples of it.
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Any multiple of --
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00:12:01,440 --> 00:12:04,200
I could multiply
this by anything.
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I might as well call it, I could
say, C, some multiple of this.
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00:12:08,430 --> 00:12:10,380
So let me --
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00:12:10,380 --> 00:12:14,890
so X could be any
multiple of this one.
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00:12:14,890 --> 00:12:17,700
OK, that, that
describes now a line,
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an infinitely long line
in four dimensional space.
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00:12:23,430 --> 00:12:26,720
But -- which is
in the null space.
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Is that the whole null space?
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00:12:29,390 --> 00:12:30,460
No.
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00:12:30,460 --> 00:12:34,220
I've got two free
variables here.
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00:12:34,220 --> 00:12:37,330
I made this choice, one and
zero, for the free variables,
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00:12:37,330 --> 00:12:40,090
but I could have
made another choice.
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00:12:40,090 --> 00:12:44,420
Let me make the other
choice zero and one.
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00:12:44,420 --> 00:12:47,140
You see my system.
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00:12:47,140 --> 00:12:48,580
So let me repeat the system.
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00:12:48,580 --> 00:12:54,087
This is the algorithm that
you, you just learned to do.
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Do elimination.
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Decide which are
the pivot columns
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and which are the free columns.
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00:13:02,420 --> 00:13:05,600
That tells you that, that
variables one and three
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00:13:05,600 --> 00:13:09,760
are pivot variables, two
and four are free variables.
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00:13:09,760 --> 00:13:14,800
Then those free variables,
you assign them --
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00:13:14,800 --> 00:13:17,870
you give one of them the value
one and the others the value
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00:13:17,870 --> 00:13:22,070
zero -- in this case, we
had a one and a zero --
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00:13:22,070 --> 00:13:24,080
and complete the solution.
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00:13:24,080 --> 00:13:28,550
And you do -- you give the other
one the value one and zero.
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And now complete the solution.
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So let's complete that solution.
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00:13:32,310 --> 00:13:35,790
I'm looking for a vector
in the null space,
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and it's absolutely going to
be different from this guy,
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because, because any
multiple of that zero
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is never going to give the one.
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So I have somebody
new in the null space,
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and let me finish it
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out.
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What's x3 here?
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00:13:50,140 --> 00:13:52,360
So we're going by
back substitution,
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looking at this equation.
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00:13:53,790 --> 00:13:59,680
Now x4 we've changed, we're
doing the other possibility,
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00:13:59,680 --> 00:14:02,870
where x2 is zero and x4 is one.
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00:14:02,870 --> 00:14:06,590
So x3 will happen
to be minus two.
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00:14:06,590 --> 00:14:10,510
And now what do I get
for that first equation?
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00:14:10,510 --> 00:14:12,420
x1 -- let's see.
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00:14:12,420 --> 00:14:19,430
Two x3s is minus four plus
two -- do I get a two there?
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00:14:19,430 --> 00:14:20,270
Perhaps, yeah.
248
00:14:20,270 --> 00:14:25,500
Two for x1, minus four, and two.
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00:14:25,500 --> 00:14:26,020
OK.
250
00:14:26,020 --> 00:14:28,360
That's in the null space.
251
00:14:28,360 --> 00:14:32,220
What does that say
about the columns?
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00:14:32,220 --> 00:14:37,320
That says that two of
this column minus two
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00:14:37,320 --> 00:14:41,900
of this column plus this column
gives zero, which it does.
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00:14:41,900 --> 00:14:46,090
Two of that minus two
of that and one of that
255
00:14:46,090 --> 00:14:47,520
gives the zero column.
256
00:14:47,520 --> 00:14:52,420
OK, now, now I've found another
vector in the null space.
257
00:14:52,420 --> 00:14:55,880
Now we're ready to tell
me the whole null space.
258
00:14:55,880 --> 00:15:00,160
What are all the
solutions to Ax=0?
259
00:15:00,160 --> 00:15:05,660
I've got this guy
and when I have him,
260
00:15:05,660 --> 00:15:10,660
what else is, goes into the
null space along with that?
261
00:15:10,660 --> 00:15:13,760
These are my two
special solutions.
262
00:15:13,760 --> 00:15:14,710
I call them special --
263
00:15:14,710 --> 00:15:16,250
I just invented that name.
264
00:15:16,250 --> 00:15:18,110
Special solutions.
265
00:15:18,110 --> 00:15:21,180
What's special about them
is the special numbers
266
00:15:21,180 --> 00:15:27,360
I gave to the free variables,
the values zero and one
267
00:15:27,360 --> 00:15:29,300
for the free variables.
268
00:15:29,300 --> 00:15:32,560
I could have given the
free variables any values
269
00:15:32,560 --> 00:15:35,440
and got vectors
in the null space.
270
00:15:35,440 --> 00:15:40,210
But this was a good way to be
sure I got t- got everybody.
271
00:15:40,210 --> 00:15:45,480
OK, so once I have him, I
also have any multiple, right?
272
00:15:45,480 --> 00:15:48,700
So I could take any
multiple of that
273
00:15:48,700 --> 00:15:50,910
and it's in the null space.
274
00:15:50,910 --> 00:15:52,150
And now what else --
275
00:15:52,150 --> 00:15:55,090
I left a little space for what?
276
00:15:55,090 --> 00:15:58,910
What -- a plus sign.
277
00:15:58,910 --> 00:16:00,580
I can take any combination.
278
00:16:00,580 --> 00:16:04,050
Here is a line of vectors
in the null space.
279
00:16:04,050 --> 00:16:06,110
A bunch of solutions.
280
00:16:06,110 --> 00:16:09,540
Would you rather I say in the
null space or would you rather
281
00:16:09,540 --> 00:16:13,330
I say, OK, I'm solving Ax=0?
282
00:16:13,330 --> 00:16:15,545
Well, really I'm solving Ux=0.
283
00:16:19,540 --> 00:16:23,290
Well, OK, let me put in
that crucial plus sign.
284
00:16:23,290 --> 00:16:29,420
I'm taking all the combinations
of my two special solutions.
285
00:16:29,420 --> 00:16:32,140
That's my conclusion there.
286
00:16:32,140 --> 00:16:37,300
The null space contains,
contains exactly
287
00:16:37,300 --> 00:16:42,270
all the combinations of
the special solutions.
288
00:16:42,270 --> 00:16:46,020
And how many special
solutions are there?
289
00:16:46,020 --> 00:16:49,650
There's one for
every free variable.
290
00:16:49,650 --> 00:16:51,330
And how many free
variables are there?
291
00:16:51,330 --> 00:16:54,860
Oh, I mean, we can see
all the whole picture now.
292
00:16:54,860 --> 00:16:59,830
If the rank R was
two, this is the,
293
00:16:59,830 --> 00:17:04,260
this is the number of
pivot variables, right,
294
00:17:04,260 --> 00:17:05,569
because it counted the pivots.
295
00:17:08,800 --> 00:17:11,960
So how many free variables?
296
00:17:11,960 --> 00:17:14,970
Well, you know it's two, right?
297
00:17:14,970 --> 00:17:20,260
What is it in -- for a matrix
that's m rows, n columns,
298
00:17:20,260 --> 00:17:25,010
n variables that
means, with rank r?
299
00:17:25,010 --> 00:17:28,420
How many free variables
have we got left?
300
00:17:28,420 --> 00:17:34,440
If r of the variables are
pivot variables, we have n-r --
301
00:17:34,440 --> 00:17:38,400
in this case four minus
two -- free variables.
302
00:17:38,400 --> 00:17:53,390
Do you see that first of all
we get clean answers here?
303
00:17:53,390 --> 00:17:59,300
We get r pivot variables -- so
there really were r equations
304
00:17:59,300 --> 00:18:00,170
here.
305
00:18:00,170 --> 00:18:01,990
There looked like
three equations,
306
00:18:01,990 --> 00:18:05,730
but there were really only
two independent equations.
307
00:18:05,730 --> 00:18:11,820
And there were n-r variables
that we could choose freely,
308
00:18:11,820 --> 00:18:16,400
and we gave them those
special zero one values,
309
00:18:16,400 --> 00:18:19,180
and we got the
special solutions.
310
00:18:19,180 --> 00:18:19,680
OK.
311
00:18:22,260 --> 00:18:25,710
For me -- we could
stop at that point.
312
00:18:25,710 --> 00:18:28,580
That gives you a
complete algorithm
313
00:18:28,580 --> 00:18:34,690
for finding all the
solutions to A x equals zero.
314
00:18:34,690 --> 00:18:35,190
OK.
315
00:18:38,300 --> 00:18:41,720
Again, you do elimination --
316
00:18:41,720 --> 00:18:46,050
going onward when a column,
when there's nothing
317
00:18:46,050 --> 00:18:50,570
to be done on one column,
you just continue.
318
00:18:50,570 --> 00:18:55,360
There's this number r, the
number of pivots, is crucial,
319
00:18:55,360 --> 00:19:01,150
and it leaves n-r free
variables, which you
320
00:19:01,150 --> 00:19:03,100
give values zero and one to.
321
00:19:03,100 --> 00:19:05,390
I would like to
take one more step.
322
00:19:08,210 --> 00:19:12,020
I would like to clean up
this matrix even more.
323
00:19:12,020 --> 00:19:14,930
So now I'm going to go
to -- this is in its,
324
00:19:14,930 --> 00:19:20,420
this is in echelon form,
upper triangular if you like.
325
00:19:20,420 --> 00:19:25,660
I want to go one more step to
make it as good as it can be.
326
00:19:25,660 --> 00:19:31,310
OK, so now I'm going to speak
about the reduced row echelon
327
00:19:31,310 --> 00:19:32,200
form.
328
00:19:32,200 --> 00:19:35,540
OK, so now I'm going to speak
about the matrix R, which is
329
00:19:35,540 --> 00:19:44,680
the reduced row echelon form.
330
00:19:44,680 --> 00:19:46,700
So what does that mean?
331
00:19:46,700 --> 00:19:49,470
That means I just --
332
00:19:49,470 --> 00:19:52,370
I can, I can work harder on U.
333
00:19:52,370 --> 00:19:55,310
So let me start, let
me suppose I got as far
334
00:19:55,310 --> 00:19:58,125
as U, which was good.
335
00:20:08,230 --> 00:20:11,450
Notice how that row
of zeros appeared.
336
00:20:11,450 --> 00:20:15,360
I didn't comment on
that, but I should have.
337
00:20:15,360 --> 00:20:20,630
That row of zeros up here
is because row three was
338
00:20:20,630 --> 00:20:23,060
a combination of
rows one and two,
339
00:20:23,060 --> 00:20:27,390
and elimination
discovered that fact.
340
00:20:27,390 --> 00:20:32,610
When we get a row of zeros,
that's telling us that the --
341
00:20:32,610 --> 00:20:38,580
original row that was there was
a combination of other rows,
342
00:20:38,580 --> 00:20:42,630
and elimination knocked it out.
343
00:20:42,630 --> 00:20:44,710
OK, so we got this far.
344
00:20:44,710 --> 00:20:47,260
Now how can I clean
that up further?
345
00:20:47,260 --> 00:20:51,430
I can do, elimination upwards.
346
00:20:51,430 --> 00:20:54,390
I can get zero above the pivots.
347
00:20:54,390 --> 00:20:58,740
So this reduced row
echelon form has zeros
348
00:20:58,740 --> 00:21:05,745
above and below the pivots.
349
00:21:08,530 --> 00:21:11,460
So let me do that.
350
00:21:11,460 --> 00:21:14,940
So now I'll subtract one
of this from the row above.
351
00:21:14,940 --> 00:21:20,150
That will leave a zero
and a minus two in there.
352
00:21:20,150 --> 00:21:22,540
And that's good.
353
00:21:26,670 --> 00:21:30,710
OK, and I can clean it
up even one more step.
354
00:21:30,710 --> 00:21:33,400
I can make the pivots --
355
00:21:33,400 --> 00:21:37,300
the pivots I'm going to make
equal to one, because I can
356
00:21:37,300 --> 00:21:41,380
divide equation
two by the pivot.
357
00:21:41,380 --> 00:21:44,110
That won't change the solutions.
358
00:21:44,110 --> 00:21:45,940
So let me do that.
359
00:21:45,940 --> 00:21:46,870
And then I really --
360
00:21:46,870 --> 00:21:47,910
I'm ready to stop.
361
00:21:47,910 --> 00:21:53,230
One, two, zero, minus
two, zero, zero, one, two.
362
00:21:53,230 --> 00:21:58,000
I divided the second
equation by two,
363
00:21:58,000 --> 00:22:05,560
because now I have a one in
the pivot and zeros below.
364
00:22:05,560 --> 00:22:06,060
OK.
365
00:22:06,060 --> 00:22:14,790
This is my matrix R.
366
00:22:14,790 --> 00:22:17,810
I guess I'm hoping
that you could now
367
00:22:17,810 --> 00:22:21,710
execute the whole algorithm.
368
00:22:21,710 --> 00:22:27,518
Matlab will do it immediately
with the command --
369
00:22:30,870 --> 00:22:34,490
reduced row echelon form of A.
370
00:22:34,490 --> 00:22:37,630
So if I input that
original matrix A
371
00:22:37,630 --> 00:22:43,360
and then I write, then I type
that command, press return,
372
00:22:43,360 --> 00:22:46,210
that matrix will appear.
373
00:22:46,210 --> 00:22:49,140
That's the reduced
row echelon form,
374
00:22:49,140 --> 00:23:00,140
and it's got all the
information as clear as can be.
375
00:23:00,140 --> 00:23:01,890
What, what information
has it got?
376
00:23:01,890 --> 00:23:04,000
Well, of course it
immediately tells me
377
00:23:04,000 --> 00:23:08,530
the pivot rows, pivot
rows, one and two,
378
00:23:08,530 --> 00:23:11,490
pivot columns, one and three.
379
00:23:11,490 --> 00:23:15,240
And in fact it's got the
identity matrix in there,
380
00:23:15,240 --> 00:23:18,820
It's, it's got zeros above
and below the pivots, right?
381
00:23:18,820 --> 00:23:22,460
and the pivots are one,
so it's, so it's got a --
382
00:23:22,460 --> 00:23:30,180
so notice the two by two
identity matrix that's sitting
383
00:23:30,180 --> 00:23:33,370
in the pivot rows
and pivot columns.
384
00:23:33,370 --> 00:23:42,710
it's I in the pivot
rows and columns.
385
00:23:46,590 --> 00:23:48,930
It's got zero rows below.
386
00:23:52,120 --> 00:23:56,580
Those are always indicating
that original rows were,
387
00:23:56,580 --> 00:23:58,740
were combinations of other rows.
388
00:23:58,740 --> 00:24:02,100
So we really only
had two rows there.
389
00:24:02,100 --> 00:24:05,740
And now it also -- so
there's the identity.
390
00:24:05,740 --> 00:24:10,740
Now it's also got
its free columns.
391
00:24:10,740 --> 00:24:17,160
And, they're cleaned
up as much as possible.
392
00:24:17,160 --> 00:24:21,720
Actually, actually it's now
so cleaned up that the special
393
00:24:21,720 --> 00:24:26,070
solutions, I can read off --
you remember I went through
394
00:24:26,070 --> 00:24:30,220
the steps of computing this --
395
00:24:30,220 --> 00:24:32,760
doing back substitution?
396
00:24:32,760 --> 00:24:37,390
Let me, let me,
instead of that system,
397
00:24:37,390 --> 00:24:39,600
let me take this
improved system.
398
00:24:39,600 --> 00:24:43,850
So I'm going to use
these numbers, right.
399
00:24:43,850 --> 00:24:45,850
In these equations,
what did I do?
400
00:24:45,850 --> 00:24:52,820
I divided this equation
by two and, oh yeah
401
00:24:52,820 --> 00:24:54,540
and I had subtracted
two of this,
402
00:24:54,540 --> 00:24:58,290
which knocked out this guy
and made that a minus sign.
403
00:24:58,290 --> 00:25:01,500
Is that what --
404
00:25:01,500 --> 00:25:04,570
I've now written Rx equals zero.
405
00:25:10,350 --> 00:25:14,570
Now I guess I'm hoping everybody
in this room understands
406
00:25:14,570 --> 00:25:19,150
the solutions to the
original A x equals zero,
407
00:25:19,150 --> 00:25:23,050
the midway, halfway,
U x equals zero,
408
00:25:23,050 --> 00:25:27,980
and the final R x equals
zero are all the same.
409
00:25:27,980 --> 00:25:30,900
Because going from one
of those to another one
410
00:25:30,900 --> 00:25:33,330
I didn't mess up.
411
00:25:33,330 --> 00:25:36,280
I just multiplied
equations and subtracted
412
00:25:36,280 --> 00:25:39,840
from other equations,
which I'm allowed to do.
413
00:25:39,840 --> 00:25:40,340
OK.
414
00:25:40,340 --> 00:25:47,530
But my point is that now
if I do this free variables
415
00:25:47,530 --> 00:25:52,410
and back substitution, it's
just, the numbers are there.
416
00:25:52,410 --> 00:26:01,040
When I let x -- so in this guy,
I let x2 be one and x4 be zero.
417
00:26:01,040 --> 00:26:04,380
I, I guess, what I seeing here?
418
00:26:04,380 --> 00:26:07,070
Let me, let me sort of
separate this out here.
419
00:26:07,070 --> 00:26:12,100
I'm seeing in the pivot,
in the pivot columns,
420
00:26:12,100 --> 00:26:16,910
if I, if I put the pivot
columns here, I'm seeing those.
421
00:26:16,910 --> 00:26:21,970
And I'm -- in the free
columns I'm seeing --
422
00:26:21,970 --> 00:26:23,570
what I seeing in
the free columns?
423
00:26:23,570 --> 00:26:29,030
A two, zero in that first
free column, the x2 column,
424
00:26:29,030 --> 00:26:33,920
and a minus two, two
in the fourth column,
425
00:26:33,920 --> 00:26:36,010
the other free column.
426
00:26:36,010 --> 00:26:41,750
And the row of zeros below,
which of course have no --
427
00:26:41,750 --> 00:26:43,860
that equation is
zero equals zero.
428
00:26:43,860 --> 00:26:46,080
That's satisfied.
429
00:26:46,080 --> 00:26:48,530
Here's my point.
430
00:26:48,530 --> 00:26:51,220
That when I do
back substitution,
431
00:26:51,220 --> 00:26:55,700
these numbers are
exactly what shows up --
432
00:26:55,700 --> 00:26:58,480
oh, their signs get switched.
433
00:26:58,480 --> 00:27:01,550
I was going to say those
numbers, two, minus two, zero,
434
00:27:01,550 --> 00:27:06,240
two, can I just circle the
-- this is the free part
435
00:27:06,240 --> 00:27:07,120
of the matrix.
436
00:27:07,120 --> 00:27:10,220
This is the identity part.
437
00:27:10,220 --> 00:27:14,710
This is the free part,
maybe I'll call it F.
438
00:27:14,710 --> 00:27:18,650
This, of course, I call I,
because it's the identity.
439
00:27:18,650 --> 00:27:25,020
The free part is a, I mean, I'm
just doing back substitution.
440
00:27:25,020 --> 00:27:29,340
And those free numbers will
show up, with a minus sign,
441
00:27:29,340 --> 00:27:32,040
because they pop onto the
other side of the equation --
442
00:27:32,040 --> 00:27:35,870
so I see minus two, zero,
and I see two, minus two.
443
00:27:39,670 --> 00:27:41,310
So that wasn't magic.
444
00:27:41,310 --> 00:27:43,540
It had to happen.
445
00:27:43,540 --> 00:27:48,240
Let me, show you
clearly why it happened.
446
00:27:48,240 --> 00:27:50,340
OK, so that's --
447
00:27:50,340 --> 00:27:53,290
this is what I'm
interested in here.
448
00:27:53,290 --> 00:27:59,410
And now let me, let me just,
like, do it, do it for --
449
00:27:59,410 --> 00:28:04,530
let's suppose we've,
we've got to --
450
00:28:07,750 --> 00:28:11,940
let's suppose we've got
this system already in,
451
00:28:11,940 --> 00:28:18,200
in rref form.
452
00:28:18,200 --> 00:28:22,580
So my matrix R is --
what does it look like?
453
00:28:22,580 --> 00:28:25,130
OK, and I'll --
454
00:28:25,130 --> 00:28:30,570
let me pretend that the
pivot columns come first
455
00:28:30,570 --> 00:28:33,220
and then whatever's
in the free columns.
456
00:28:33,220 --> 00:28:37,970
And there might be
some zero rows below.
457
00:28:37,970 --> 00:28:40,600
There's a typical --
458
00:28:40,600 --> 00:28:45,750
a pretty typical reduced
row echelon form.
459
00:28:49,450 --> 00:28:51,510
You see what's typical.
460
00:28:51,510 --> 00:28:55,430
It's got -- this is r by r.
461
00:28:55,430 --> 00:28:57,950
This is r pivot rows.
462
00:29:02,610 --> 00:29:06,050
This is r pivot columns.
463
00:29:09,780 --> 00:29:16,290
And here are n-r free columns.
464
00:29:16,290 --> 00:29:17,480
OK.
465
00:29:17,480 --> 00:29:21,650
Tell me what are the
special solutions?
466
00:29:21,650 --> 00:29:23,350
What are the --
467
00:29:23,350 --> 00:29:24,060
what's x?
468
00:29:24,060 --> 00:29:27,440
If I want to solve
R x equals zero --
469
00:29:27,440 --> 00:29:31,170
in fact, let me --
470
00:29:31,170 --> 00:29:36,110
I'm really going to, do
the whole -- since these --
471
00:29:36,110 --> 00:29:39,030
this is now block matrices,
I might as well do all
472
00:29:39,030 --> 00:29:41,300
of the special
solutions at once.
473
00:29:41,300 --> 00:29:44,740
So I want to solve
R x equals zero,
474
00:29:44,740 --> 00:29:49,660
and I'll have some
special solutions.
475
00:29:49,660 --> 00:29:52,830
Let me, actually --
476
00:29:52,830 --> 00:29:54,780
can I do them all at once?
477
00:29:54,780 --> 00:30:00,750
I'm going to create a
null space matrix, OK.
478
00:30:00,750 --> 00:30:01,790
A matrix.
479
00:30:04,970 --> 00:30:13,130
Its, its, its columns
are the special --
480
00:30:13,130 --> 00:30:15,085
the columns are the
special solutions.
481
00:30:19,080 --> 00:30:21,070
This is, I'm making
it sound harder,
482
00:30:21,070 --> 00:30:22,670
it's going to be totally easy.
483
00:30:22,670 --> 00:30:25,800
N will be this
null space matrix.
484
00:30:25,800 --> 00:30:31,070
I want R N to be
the zero matrix.
485
00:30:31,070 --> 00:30:33,830
These columns of N are
supposed to multipl-
486
00:30:33,830 --> 00:30:37,300
to get multiplied by R
and give zero columns.
487
00:30:37,300 --> 00:30:40,100
So what N will do the job?
488
00:30:40,100 --> 00:30:41,450
Let me put --
489
00:30:41,450 --> 00:30:45,290
I'm going to put the identity
in the free variable part
490
00:30:45,290 --> 00:30:54,020
and then minus F will show up
in the pivot variables, just
491
00:30:54,020 --> 00:30:55,980
the way it did in that example.
492
00:30:55,980 --> 00:30:58,970
There we had the identity and F.
493
00:30:58,970 --> 00:31:02,160
Here -- in the special solution.
494
00:31:02,160 --> 00:31:05,400
So these columns are --
there's the matrix of special
495
00:31:05,400 --> 00:31:06,430
solutions.
496
00:31:06,430 --> 00:31:09,460
And actually, there -- so
there's a Matlab command
497
00:31:09,460 --> 00:31:14,360
or a teaching code
command, NULL --
498
00:31:14,360 --> 00:31:19,460
N equal, so this is the --
499
00:31:19,460 --> 00:31:24,710
produces the null basis, the
null space matrix, NULL of A,
500
00:31:24,710 --> 00:31:26,230
and there it is.
501
00:31:30,450 --> 00:31:33,630
And how does that
command actually work?
502
00:31:33,630 --> 00:31:38,590
It uses Matlab to
compute R, then
503
00:31:38,590 --> 00:31:43,100
it picks out the pivot
variables, the free variables,
504
00:31:43,100 --> 00:31:47,980
puts, puts ones and zeros
in for the free variables,
505
00:31:47,980 --> 00:31:51,850
and copies out the
pivot variables.
506
00:31:51,850 --> 00:31:54,760
It, it does back substitution,
but back substitution
507
00:31:54,760 --> 00:31:57,180
for this system
is totally simple.
508
00:31:57,180 --> 00:32:00,070
What is this system?
509
00:32:00,070 --> 00:32:03,030
R x equals zero.
510
00:32:03,030 --> 00:32:11,620
So this is R is I F, and
x is the pivot variables
511
00:32:11,620 --> 00:32:18,240
and the free variables, and
it's supposed to give zero.
512
00:32:18,240 --> 00:32:20,240
So what does that mean?
513
00:32:20,240 --> 00:32:24,960
That means that the
pivot variables plus F
514
00:32:24,960 --> 00:32:28,640
times the free
variables give zero.
515
00:32:28,640 --> 00:32:31,950
So let me put F times the free
variables on the other side.
516
00:32:31,950 --> 00:32:37,950
I get minus F times
the free variables.
517
00:32:37,950 --> 00:32:43,530
There's my, equation,
as simple as it can be.
518
00:32:43,530 --> 00:32:45,840
That's what back
substitution comes
519
00:32:45,840 --> 00:32:49,280
to when I've reduced and
reduced and reduced this system
520
00:32:49,280 --> 00:32:52,090
to the, to the best form, OK.
521
00:32:52,090 --> 00:32:56,680
And, then if the
free variables, I
522
00:32:56,680 --> 00:33:00,070
make this special
choice of the identity,
523
00:33:00,070 --> 00:33:02,570
then the pivot variables are
524
00:33:02,570 --> 00:33:08,710
minus F. OK, can I
do, another example?
525
00:33:08,710 --> 00:33:10,180
Could you do another example?
526
00:33:10,180 --> 00:33:12,300
Can I -- let me just
take another matrix
527
00:33:12,300 --> 00:33:17,330
and, and let's go through
this algorithm once more, OK.
528
00:33:17,330 --> 00:33:19,180
Here we go.
529
00:33:19,180 --> 00:33:25,100
Here's a blackboard
for another matrix, OK.
530
00:33:25,100 --> 00:33:31,180
So I'll call the matrix A
again, but let me make it --
531
00:33:31,180 --> 00:33:33,680
yeah, how big shall
we make it this time?
532
00:33:36,480 --> 00:33:38,620
Why don't I do this?
533
00:33:38,620 --> 00:33:39,920
Just for the heck of it.
534
00:33:39,920 --> 00:33:44,860
Let me take the transpose of
this A and see what happens to
535
00:33:44,860 --> 00:33:45,670
that.
536
00:33:45,670 --> 00:33:55,820
Two four six eight and
three six eight ten.
537
00:34:00,250 --> 00:34:06,380
Before we do the calculations,
tell me what's coming?
538
00:34:06,380 --> 00:34:12,170
How many pivot variables
do you expect here?
539
00:34:12,170 --> 00:34:16,900
How many columns are
going to have pivots?
540
00:34:16,900 --> 00:34:22,060
How many -- we have three
columns in that matrix,
541
00:34:22,060 --> 00:34:25,739
but are we going to, are we
going to have three pivots?
542
00:34:25,739 --> 00:34:31,429
No, because this third columns
is the sum of the first two
543
00:34:31,429 --> 00:34:32,100
columns.
544
00:34:32,100 --> 00:34:38,350
I'm totally expecting, totally
expecting that these will be
545
00:34:38,350 --> 00:34:41,000
pivot columns --
546
00:34:41,000 --> 00:34:45,949
because they're independent,
but that this third guy,
547
00:34:45,949 --> 00:34:49,659
the third column, which is
dependent on the first two,
548
00:34:49,659 --> 00:34:52,219
is going to be a free column.
549
00:34:52,219 --> 00:34:54,889
Elimination better
discover that.
550
00:34:54,889 --> 00:34:58,270
And elimination will
also straighten out
551
00:34:58,270 --> 00:35:05,170
the rows, dependent rows
and some independent rows.
552
00:35:05,170 --> 00:35:09,510
What's the, what's the row
reduced echelon form for this?
553
00:35:09,510 --> 00:35:11,970
Let's just do it, OK.
554
00:35:11,970 --> 00:35:16,710
So, so that's the first pivot.
555
00:35:16,710 --> 00:35:20,360
Two times that away from
that gives me a row of zeros.
556
00:35:20,360 --> 00:35:24,970
Two times that away from
that gives me a zero two two.
557
00:35:24,970 --> 00:35:28,865
And two times that away from
that gives me a zero four four.
558
00:35:32,300 --> 00:35:35,750
OK, first column is straight.
559
00:35:35,750 --> 00:35:37,780
First variable is
a pivot variable.
560
00:35:37,780 --> 00:35:39,010
No problem.
561
00:35:39,010 --> 00:35:40,370
On to the second column.
562
00:35:40,370 --> 00:35:43,880
I look at the second
pivot, it's a zero.
563
00:35:43,880 --> 00:35:45,430
I look below it.
564
00:35:45,430 --> 00:35:46,740
There's a two.
565
00:35:46,740 --> 00:35:47,890
OK, I do a row exchange.
566
00:35:50,530 --> 00:35:53,630
So this zero is now there.
567
00:35:53,630 --> 00:35:58,710
I now have a perfectly
good pivot, and I use it.
568
00:35:58,710 --> 00:36:03,030
OK, and I subtract two of
that row away from this row.
569
00:36:03,030 --> 00:36:05,630
All right if I do it like that?
570
00:36:05,630 --> 00:36:08,310
I've got to the form U now.
571
00:36:08,310 --> 00:36:10,460
This was my A.
572
00:36:10,460 --> 00:36:17,260
Now there's my U.
I can see now --
573
00:36:17,260 --> 00:36:18,630
I have to stop, right?
574
00:36:18,630 --> 00:36:20,340
I would go on to
the third column.
575
00:36:20,340 --> 00:36:22,650
I should have tried.
576
00:36:22,650 --> 00:36:24,120
I quit, but without trying.
577
00:36:24,120 --> 00:36:25,440
I shouldn't have done that.
578
00:36:25,440 --> 00:36:28,810
On to the third column,
look at the pivot position.
579
00:36:28,810 --> 00:36:30,100
It's got a zero in it.
580
00:36:30,100 --> 00:36:32,300
Look below, all zeros.
581
00:36:32,300 --> 00:36:35,660
Now I'm entitled to stop, OK.
582
00:36:35,660 --> 00:36:37,445
So the rank is two again.
583
00:36:45,860 --> 00:36:48,440
What about the null space?
584
00:36:48,440 --> 00:36:52,450
How many special solutions
are there this time?
585
00:36:52,450 --> 00:36:56,990
How many special
solutions for this matrix?
586
00:36:56,990 --> 00:36:59,600
I've got -- and which are the
free variables and which are
587
00:36:59,600 --> 00:37:01,400
the pivot variables and so on?
588
00:37:01,400 --> 00:37:04,610
Pivot columns, I've
got two pivot columns,
589
00:37:04,610 --> 00:37:06,890
and that's no accident.
590
00:37:06,890 --> 00:37:11,200
The number of pivot columns
for a matrix A, that's
591
00:37:11,200 --> 00:37:16,120
a great fact, that the
number of pivot columns for A
592
00:37:16,120 --> 00:37:19,250
and A transpose are the same.
593
00:37:19,250 --> 00:37:21,680
And then I have a free column.
594
00:37:21,680 --> 00:37:24,240
There's a free column.
595
00:37:24,240 --> 00:37:30,330
One free column, because the
count is three minus two.
596
00:37:30,330 --> 00:37:33,990
Three minus two gives
me one free column.
597
00:37:37,760 --> 00:37:47,410
OK, so now let me solve,
what's in the null space.
598
00:37:47,410 --> 00:37:49,390
OK, so how do I --
599
00:37:49,390 --> 00:37:50,530
let's see.
600
00:37:50,530 --> 00:37:52,800
These vectors have length three.
601
00:37:52,800 --> 00:37:54,480
They only have three components.
602
00:37:54,480 --> 00:37:58,200
I'm making too much space
for the, to write x.
603
00:37:58,200 --> 00:38:03,320
x has just got three
components, and what are they?
604
00:38:03,320 --> 00:38:06,340
I'm looking for the null space.
605
00:38:09,660 --> 00:38:12,430
OK, so how do I start?
606
00:38:12,430 --> 00:38:17,710
I give the free variable
some convenient value.
607
00:38:17,710 --> 00:38:20,290
And what's that?
608
00:38:20,290 --> 00:38:23,000
I set it to one.
609
00:38:23,000 --> 00:38:24,970
I set the free variable to one.
610
00:38:24,970 --> 00:38:28,520
If I set the free
variable to zero and solve
611
00:38:28,520 --> 00:38:33,300
for the pivot variables, I'll
get all zeros: no progress.
612
00:38:33,300 --> 00:38:36,430
But by setting the
free variable to one --
613
00:38:36,430 --> 00:38:39,680
you see w- my two
equations now are --
614
00:38:39,680 --> 00:38:45,230
my equations are x1+
2x2+ 3 x3 is zero,
615
00:38:45,230 --> 00:38:47,200
that's my first equation.
616
00:38:47,200 --> 00:38:51,130
And my second equation is
now 2x2+2x3 equals zero.
617
00:38:51,130 --> 00:38:56,110
And, OK.
618
00:38:56,110 --> 00:39:02,400
So if x3 is one,
then x2 is minus one.
619
00:39:02,400 --> 00:39:07,430
And if x3 is one and x2 is
minus one, then maybe x1
620
00:39:07,430 --> 00:39:08,510
is minus one.
621
00:39:12,080 --> 00:39:14,360
And actually I go
back to check now.
622
00:39:14,360 --> 00:39:17,410
I don't, like --
623
00:39:17,410 --> 00:39:19,410
I did a quick
calculation mentally.
624
00:39:19,410 --> 00:39:21,570
Can I mentally do a quick check?
625
00:39:21,570 --> 00:39:24,180
That matrix, that
solution x says
626
00:39:24,180 --> 00:39:28,710
that minus this column minus
this column plus this one
627
00:39:28,710 --> 00:39:31,560
is the zero column.
628
00:39:31,560 --> 00:39:32,860
And it is.
629
00:39:32,860 --> 00:39:35,510
Minus that minus that
plus that is zero.
630
00:39:35,510 --> 00:39:37,720
So that's in the null space.
631
00:39:37,720 --> 00:39:40,790
And now you can tell me what
else is in the null space.
632
00:39:40,790 --> 00:39:44,080
What's, what's the
whole null space now?
633
00:39:44,080 --> 00:39:46,570
I multiply by C, right.
634
00:39:46,570 --> 00:39:51,720
The whole null space is a line.
635
00:39:51,720 --> 00:39:52,970
So that's the description.
636
00:39:52,970 --> 00:39:57,520
You know, if I ask you on a
homework or a quiz or the final
637
00:39:57,520 --> 00:40:01,630
what -- give me, describe,
tell me the null space,
638
00:40:01,630 --> 00:40:04,410
find the null space
of this matrix,
639
00:40:04,410 --> 00:40:07,340
you can take those steps.
640
00:40:07,340 --> 00:40:10,790
And that's the answer
I'm looking for.
641
00:40:10,790 --> 00:40:15,190
And I'm looking for that C
too, because that's telling me
642
00:40:15,190 --> 00:40:18,360
that you're remembering that
it's a whole space and not
643
00:40:18,360 --> 00:40:20,640
just one vector.
644
00:40:20,640 --> 00:40:23,960
Later I will ask you for a
basis for the null space.
645
00:40:23,960 --> 00:40:26,850
Then I just want this vector.
646
00:40:26,850 --> 00:40:28,840
But if I ask for the
whole null space,
647
00:40:28,840 --> 00:40:31,510
then there's the whole
line through that vector.
648
00:40:31,510 --> 00:40:36,340
OK, now one more natural thing
to do with this example, right,
649
00:40:36,340 --> 00:40:43,300
is keep going to the
reduced matrix, R.
650
00:40:43,300 --> 00:40:46,180
So can I push onwards to R?
651
00:40:46,180 --> 00:40:49,340
That should be quick,
but let's just practice.
652
00:40:49,340 --> 00:40:54,000
Let me keep going to R.
OK, so what do I do here?
653
00:40:54,000 --> 00:40:55,970
I subtract --
654
00:40:55,970 --> 00:40:57,830
I clear out above
the pivot, so I
655
00:40:57,830 --> 00:41:02,870
subtract that from that, that's
one zero one is left, right?
656
00:41:02,870 --> 00:41:04,740
When I subtracted
this row from this
657
00:41:04,740 --> 00:41:07,390
it produced a zero
above this pivot.
658
00:41:07,390 --> 00:41:12,070
And now I want that
pivot to be a one.
659
00:41:12,070 --> 00:41:18,160
So for the R matrix, I'll
divide this equation by two,
660
00:41:18,160 --> 00:41:23,270
and of course these zero, zeros
are great, they don't change.
661
00:41:23,270 --> 00:41:24,700
There's R.
662
00:41:24,700 --> 00:41:27,370
That's R.
663
00:41:27,370 --> 00:41:28,430
You see what R is?
664
00:41:28,430 --> 00:41:32,100
You see the identity
matrix sitting up here?
665
00:41:32,100 --> 00:41:36,370
You see the free part
F, the F part here?
666
00:41:36,370 --> 00:41:38,370
And you see the zeros below.
667
00:41:38,370 --> 00:41:42,460
This is I F zero zero.
668
00:41:42,460 --> 00:41:45,180
And what's the x?
669
00:41:45,180 --> 00:41:48,660
The x has the identity --
670
00:41:48,660 --> 00:41:51,460
well, it's only a
single number one,
671
00:41:51,460 --> 00:41:56,090
but it's the identity matrix
in the free, in the free part.
672
00:41:56,090 --> 00:42:00,770
And what does it have
in the pivot variables?
673
00:42:00,770 --> 00:42:03,620
What did back substitution give?
674
00:42:03,620 --> 00:42:06,910
It gave minus these guys.
675
00:42:06,910 --> 00:42:11,160
You see that what this
is is any multiple of --
676
00:42:11,160 --> 00:42:14,360
this is the identity there,
and this is minus F here.
677
00:42:19,190 --> 00:42:24,710
This is our null space
matrix N for this.
678
00:42:24,710 --> 00:42:28,490
Our, our null space matrix
is the guy whose columns
679
00:42:28,490 --> 00:42:30,680
are the special solutions.
680
00:42:30,680 --> 00:42:33,570
So their free variables
have the special values
681
00:42:33,570 --> 00:42:39,510
one and, pivot
variables have minus F.
682
00:42:39,510 --> 00:42:42,210
So do you see, though,
how the minus F just
683
00:42:42,210 --> 00:42:45,540
automatically shows up
in the special solutions.
684
00:42:49,390 --> 00:42:50,389
That's it really.
685
00:42:50,389 --> 00:42:51,930
I don't think there's
anything more I
686
00:42:51,930 --> 00:42:55,980
can say about A x equals zero.
687
00:42:55,980 --> 00:42:59,640
There's more I can
say about A x equal b,
688
00:42:59,640 --> 00:43:03,080
but that'll be on Friday.
689
00:43:03,080 --> 00:43:05,370
OK, so that's, that's
the null space.
690
00:43:05,370 --> 00:43:06,920
Thanks.