WEBVTT
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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?
00:11:19.740 --> 00:11:24.360
That simply says that minus
two times the first column
00:11:24.360 --> 00:11:28.502
plus one times the second
column is the zero column.
00:11:28.502 --> 00:11:29.460
Of course that's right.
00:11:29.460 --> 00:11:33.310
Minus two of that column plus
one of that, or minus two
00:11:33.310 --> 00:11:35.580
of that plus one of that.
00:11:35.580 --> 00:11:39.620
That solution is -- that, that's
just what we saw immediately,
00:11:39.620 --> 00:11:43.690
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.
00:11:51.580 --> 00:11:58.030
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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I could multiply
this by anything.
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I might as well call it, I could
say, C, some multiple of this.
00:12:08.430 --> 00:12:10.380
So let me --
00:12:10.380 --> 00:12:14.890
so X could be any
multiple of this one.
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OK, that, that
describes now a line,
00:12:17.700 --> 00:12:23.430
an infinitely long line
in four dimensional space.
00:12:23.430 --> 00:12:26.720
But -- which is
in the null space.
00:12:26.720 --> 00:12:29.390
Is that the whole null space?
00:12:29.390 --> 00:12:30.460
No.
00:12:30.460 --> 00:12:34.220
I've got two free
variables here.
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I made this choice, one and
zero, for the free variables,
00:12:37.330 --> 00:12:40.090
but I could have
made another choice.
00:12:40.090 --> 00:12:44.420
Let me make the other
choice zero and one.
00:12:44.420 --> 00:12:47.140
You see my system.
00:12:47.140 --> 00:12:48.580
So let me repeat the system.
00:12:48.580 --> 00:12:54.087
This is the algorithm that
you, you just learned to do.
00:12:56.890 --> 00:12:59.100
Do elimination.
00:12:59.100 --> 00:13:01.060
Decide which are
the pivot columns
00:13:01.060 --> 00:13:02.420
and which are the free columns.
00:13:02.420 --> 00:13:05.600
That tells you that, that
variables one and three
00:13:05.600 --> 00:13:09.760
are pivot variables, two
and four are free variables.
00:13:09.760 --> 00:13:14.800
Then those free variables,
you assign them --
00:13:14.800 --> 00:13:17.870
you give one of them the value
one and the others the value
00:13:17.870 --> 00:13:22.070
zero -- in this case, we
had a one and a zero --
00:13:22.070 --> 00:13:24.080
and complete the solution.
00:13:24.080 --> 00:13:28.550
And you do -- you give the other
one the value one and zero.
00:13:28.550 --> 00:13:30.160
And now complete the solution.
00:13:30.160 --> 00:13:32.310
So let's complete that solution.
00:13:32.310 --> 00:13:35.790
I'm looking for a vector
in the null space,
00:13:35.790 --> 00:13:38.550
and it's absolutely going to
be different from this guy,
00:13:38.550 --> 00:13:42.350
because, because any
multiple of that zero
00:13:42.350 --> 00:13:43.920
is never going to give the one.
00:13:43.920 --> 00:13:46.050
So I have somebody
new in the null space,
00:13:46.050 --> 00:13:46.930
and let me finish it
00:13:46.930 --> 00:13:47.430
out.
00:13:47.430 --> 00:13:50.140
What's x3 here?
00:13:50.140 --> 00:13:52.360
So we're going by
back substitution,
00:13:52.360 --> 00:13:53.790
looking at this equation.
00:13:53.790 --> 00:13:59.680
Now x4 we've changed, we're
doing the other possibility,
00:13:59.680 --> 00:14:02.870
where x2 is zero and x4 is one.
00:14:02.870 --> 00:14:06.590
So x3 will happen
to be minus two.
00:14:06.590 --> 00:14:10.510
And now what do I get
for that first equation?
00:14:10.510 --> 00:14:12.420
x1 -- let's see.
00:14:12.420 --> 00:14:19.430
Two x3s is minus four plus
two -- do I get a two there?
00:14:19.430 --> 00:14:20.270
Perhaps, yeah.
00:14:20.270 --> 00:14:25.500
Two for x1, minus four, and two.
00:14:25.500 --> 00:14:26.020
OK.
00:14:26.020 --> 00:14:28.360
That's in the null space.
00:14:28.360 --> 00:14:32.220
What does that say
about the columns?
00:14:32.220 --> 00:14:37.320
That says that two of
this column minus two
00:14:37.320 --> 00:14:41.900
of this column plus this column
gives zero, which it does.
00:14:41.900 --> 00:14:46.090
Two of that minus two
of that and one of that
00:14:46.090 --> 00:14:47.520
gives the zero column.
00:14:47.520 --> 00:14:52.420
OK, now, now I've found another
vector in the null space.
00:14:52.420 --> 00:14:55.880
Now we're ready to tell
me the whole null space.
00:14:55.880 --> 00:15:00.160
What are all the
solutions to Ax=0?
00:15:00.160 --> 00:15:05.660
I've got this guy
and when I have him,
00:15:05.660 --> 00:15:10.660
what else is, goes into the
null space along with that?
00:15:10.660 --> 00:15:13.760
These are my two
special solutions.
00:15:13.760 --> 00:15:14.710
I call them special --
00:15:14.710 --> 00:15:16.250
I just invented that name.
00:15:16.250 --> 00:15:18.110
Special solutions.
00:15:18.110 --> 00:15:21.180
What's special about them
is the special numbers
00:15:21.180 --> 00:15:27.360
I gave to the free variables,
the values zero and one
00:15:27.360 --> 00:15:29.300
for the free variables.
00:15:29.300 --> 00:15:32.560
I could have given the
free variables any values
00:15:32.560 --> 00:15:35.440
and got vectors
in the null space.
00:15:35.440 --> 00:15:40.210
But this was a good way to be
sure I got t- got everybody.
00:15:40.210 --> 00:15:45.480
OK, so once I have him, I
also have any multiple, right?
00:15:45.480 --> 00:15:48.700
So I could take any
multiple of that
00:15:48.700 --> 00:15:50.910
and it's in the null space.
00:15:50.910 --> 00:15:52.150
And now what else --
00:15:52.150 --> 00:15:55.090
I left a little space for what?
00:15:55.090 --> 00:15:58.910
What -- a plus sign.
00:15:58.910 --> 00:16:00.580
I can take any combination.
00:16:00.580 --> 00:16:04.050
Here is a line of vectors
in the null space.
00:16:04.050 --> 00:16:06.110
A bunch of solutions.
00:16:06.110 --> 00:16:09.540
Would you rather I say in the
null space or would you rather
00:16:09.540 --> 00:16:13.330
I say, OK, I'm solving Ax=0?
00:16:13.330 --> 00:16:15.545
Well, really I'm solving Ux=0.
00:16:19.540 --> 00:16:23.290
Well, OK, let me put in
that crucial plus sign.
00:16:23.290 --> 00:16:29.420
I'm taking all the combinations
of my two special solutions.
00:16:29.420 --> 00:16:32.140
That's my conclusion there.
00:16:32.140 --> 00:16:37.300
The null space contains,
contains exactly
00:16:37.300 --> 00:16:42.270
all the combinations of
the special solutions.
00:16:42.270 --> 00:16:46.020
And how many special
solutions are there?
00:16:46.020 --> 00:16:49.650
There's one for
every free variable.
00:16:49.650 --> 00:16:51.330
And how many free
variables are there?
00:16:51.330 --> 00:16:54.860
Oh, I mean, we can see
all the whole picture now.
00:16:54.860 --> 00:16:59.830
If the rank R was
two, this is the,
00:16:59.830 --> 00:17:04.260
this is the number of
pivot variables, right,
00:17:04.260 --> 00:17:05.569
because it counted the pivots.
00:17:08.800 --> 00:17:11.960
So how many free variables?
00:17:11.960 --> 00:17:14.970
Well, you know it's two, right?
00:17:14.970 --> 00:17:20.260
What is it in -- for a matrix
that's m rows, n columns,
00:17:20.260 --> 00:17:25.010
n variables that
means, with rank r?
00:17:25.010 --> 00:17:28.420
How many free variables
have we got left?
00:17:28.420 --> 00:17:34.440
If r of the variables are
pivot variables, we have n-r --
00:17:34.440 --> 00:17:38.400
in this case four minus
two -- free variables.
00:17:38.400 --> 00:17:53.390
Do you see that first of all
we get clean answers here?
00:17:53.390 --> 00:17:59.300
We get r pivot variables -- so
there really were r equations
00:17:59.300 --> 00:18:00.170
here.
00:18:00.170 --> 00:18:01.990
There looked like
three equations,
00:18:01.990 --> 00:18:05.730
but there were really only
two independent equations.
00:18:05.730 --> 00:18:11.820
And there were n-r variables
that we could choose freely,
00:18:11.820 --> 00:18:16.400
and we gave them those
special zero one values,
00:18:16.400 --> 00:18:19.180
and we got the
special solutions.
00:18:19.180 --> 00:18:19.680
OK.
00:18:22.260 --> 00:18:25.710
For me -- we could
stop at that point.
00:18:25.710 --> 00:18:28.580
That gives you a
complete algorithm
00:18:28.580 --> 00:18:34.690
for finding all the
solutions to A x equals zero.
00:18:34.690 --> 00:18:35.190
OK.
00:18:38.300 --> 00:18:41.720
Again, you do elimination --
00:18:41.720 --> 00:18:46.050
going onward when a column,
when there's nothing
00:18:46.050 --> 00:18:50.570
to be done on one column,
you just continue.
00:18:50.570 --> 00:18:55.360
There's this number r, the
number of pivots, is crucial,
00:18:55.360 --> 00:19:01.150
and it leaves n-r free
variables, which you
00:19:01.150 --> 00:19:03.100
give values zero and one to.
00:19:03.100 --> 00:19:05.390
I would like to
take one more step.
00:19:08.210 --> 00:19:12.020
I would like to clean up
this matrix even more.
00:19:12.020 --> 00:19:14.930
So now I'm going to go
to -- this is in its,
00:19:14.930 --> 00:19:20.420
this is in echelon form,
upper triangular if you like.
00:19:20.420 --> 00:19:25.660
I want to go one more step to
make it as good as it can be.
00:19:25.660 --> 00:19:31.310
OK, so now I'm going to speak
about the reduced row echelon
00:19:31.310 --> 00:19:32.200
form.
00:19:32.200 --> 00:19:35.540
OK, so now I'm going to speak
about the matrix R, which is
00:19:35.540 --> 00:19:44.680
the reduced row echelon form.
00:19:44.680 --> 00:19:46.700
So what does that mean?
00:19:46.700 --> 00:19:49.470
That means I just --
00:19:49.470 --> 00:19:52.370
I can, I can work harder on U.
00:19:52.370 --> 00:19:55.310
So let me start, let
me suppose I got as far
00:19:55.310 --> 00:19:58.125
as U, which was good.
00:20:08.230 --> 00:20:11.450
Notice how that row
of zeros appeared.
00:20:11.450 --> 00:20:15.360
I didn't comment on
that, but I should have.
00:20:15.360 --> 00:20:20.630
That row of zeros up here
is because row three was
00:20:20.630 --> 00:20:23.060
a combination of
rows one and two,
00:20:23.060 --> 00:20:27.390
and elimination
discovered that fact.
00:20:27.390 --> 00:20:32.610
When we get a row of zeros,
that's telling us that the --
00:20:32.610 --> 00:20:38.580
original row that was there was
a combination of other rows,
00:20:38.580 --> 00:20:42.630
and elimination knocked it out.
00:20:42.630 --> 00:20:44.710
OK, so we got this far.
00:20:44.710 --> 00:20:47.260
Now how can I clean
that up further?
00:20:47.260 --> 00:20:51.430
I can do, elimination upwards.
00:20:51.430 --> 00:20:54.390
I can get zero above the pivots.
00:20:54.390 --> 00:20:58.740
So this reduced row
echelon form has zeros
00:20:58.740 --> 00:21:05.745
above and below the pivots.
00:21:08.530 --> 00:21:11.460
So let me do that.
00:21:11.460 --> 00:21:14.940
So now I'll subtract one
of this from the row above.
00:21:14.940 --> 00:21:20.150
That will leave a zero
and a minus two in there.
00:21:20.150 --> 00:21:22.540
And that's good.
00:21:26.670 --> 00:21:30.710
OK, and I can clean it
up even one more step.
00:21:30.710 --> 00:21:33.400
I can make the pivots --
00:21:33.400 --> 00:21:37.300
the pivots I'm going to make
equal to one, because I can
00:21:37.300 --> 00:21:41.380
divide equation
two by the pivot.
00:21:41.380 --> 00:21:44.110
That won't change the solutions.
00:21:44.110 --> 00:21:45.940
So let me do that.
00:21:45.940 --> 00:21:46.870
And then I really --
00:21:46.870 --> 00:21:47.910
I'm ready to stop.
00:21:47.910 --> 00:21:53.230
One, two, zero, minus
two, zero, zero, one, two.
00:21:53.230 --> 00:21:58.000
I divided the second
equation by two,
00:21:58.000 --> 00:22:05.560
because now I have a one in
the pivot and zeros below.
00:22:05.560 --> 00:22:06.060
OK.
00:22:06.060 --> 00:22:14.790
This is my matrix R.
00:22:14.790 --> 00:22:17.810
I guess I'm hoping
that you could now
00:22:17.810 --> 00:22:21.710
execute the whole algorithm.
00:22:21.710 --> 00:22:27.518
Matlab will do it immediately
with the command --
00:22:30.870 --> 00:22:34.490
reduced row echelon form of A.
00:22:34.490 --> 00:22:37.630
So if I input that
original matrix A
00:22:37.630 --> 00:22:43.360
and then I write, then I type
that command, press return,
00:22:43.360 --> 00:22:46.210
that matrix will appear.
00:22:46.210 --> 00:22:49.140
That's the reduced
row echelon form,
00:22:49.140 --> 00:23:00.140
and it's got all the
information as clear as can be.
00:23:00.140 --> 00:23:01.890
What, what information
has it got?
00:23:01.890 --> 00:23:04.000
Well, of course it
immediately tells me
00:23:04.000 --> 00:23:08.530
the pivot rows, pivot
rows, one and two,
00:23:08.530 --> 00:23:11.490
pivot columns, one and three.
00:23:11.490 --> 00:23:15.240
And in fact it's got the
identity matrix in there,
00:23:15.240 --> 00:23:18.820
It's, it's got zeros above
and below the pivots, right?
00:23:18.820 --> 00:23:22.460
and the pivots are one,
so it's, so it's got a --
00:23:22.460 --> 00:23:30.180
so notice the two by two
identity matrix that's sitting
00:23:30.180 --> 00:23:33.370
in the pivot rows
and pivot columns.
00:23:33.370 --> 00:23:42.710
it's I in the pivot
rows and columns.
00:23:46.590 --> 00:23:48.930
It's got zero rows below.
00:23:52.120 --> 00:23:56.580
Those are always indicating
that original rows were,
00:23:56.580 --> 00:23:58.740
were combinations of other rows.
00:23:58.740 --> 00:24:02.100
So we really only
had two rows there.
00:24:02.100 --> 00:24:05.740
And now it also -- so
there's the identity.
00:24:05.740 --> 00:24:10.740
Now it's also got
its free columns.
00:24:10.740 --> 00:24:17.160
And, they're cleaned
up as much as possible.
00:24:17.160 --> 00:24:21.720
Actually, actually it's now
so cleaned up that the special
00:24:21.720 --> 00:24:26.070
solutions, I can read off --
you remember I went through
00:24:26.070 --> 00:24:30.220
the steps of computing this --
00:24:30.220 --> 00:24:32.760
doing back substitution?
00:24:32.760 --> 00:24:37.390
Let me, let me,
instead of that system,
00:24:37.390 --> 00:24:39.600
let me take this
improved system.
00:24:39.600 --> 00:24:43.850
So I'm going to use
these numbers, right.
00:24:43.850 --> 00:24:45.850
In these equations,
what did I do?
00:24:45.850 --> 00:24:52.820
I divided this equation
by two and, oh yeah
00:24:52.820 --> 00:24:54.540
and I had subtracted
two of this,
00:24:54.540 --> 00:24:58.290
which knocked out this guy
and made that a minus sign.
00:24:58.290 --> 00:25:01.500
Is that what --
00:25:01.500 --> 00:25:04.570
I've now written Rx equals zero.
00:25:10.350 --> 00:25:14.570
Now I guess I'm hoping everybody
in this room understands
00:25:14.570 --> 00:25:19.150
the solutions to the
original A x equals zero,
00:25:19.150 --> 00:25:23.050
the midway, halfway,
U x equals zero,
00:25:23.050 --> 00:25:27.980
and the final R x equals
zero are all the same.
00:25:27.980 --> 00:25:30.900
Because going from one
of those to another one
00:25:30.900 --> 00:25:33.330
I didn't mess up.
00:25:33.330 --> 00:25:36.280
I just multiplied
equations and subtracted
00:25:36.280 --> 00:25:39.840
from other equations,
which I'm allowed to do.
00:25:39.840 --> 00:25:40.340
OK.
00:25:40.340 --> 00:25:47.530
But my point is that now
if I do this free variables
00:25:47.530 --> 00:25:52.410
and back substitution, it's
just, the numbers are there.
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.
00:26:01.040 --> 00:26:04.380
I, I guess, what I seeing here?
00:26:04.380 --> 00:26:07.070
Let me, let me sort of
separate this out here.
00:26:07.070 --> 00:26:12.100
I'm seeing in the pivot,
in the pivot columns,
00:26:12.100 --> 00:26:16.910
if I, if I put the pivot
columns here, I'm seeing those.
00:26:16.910 --> 00:26:21.970
And I'm -- in the free
columns I'm seeing --
00:26:21.970 --> 00:26:23.570
what I seeing in
the free columns?
00:26:23.570 --> 00:26:29.030
A two, zero in that first
free column, the x2 column,
00:26:29.030 --> 00:26:33.920
and a minus two, two
in the fourth column,
00:26:33.920 --> 00:26:36.010
the other free column.
00:26:36.010 --> 00:26:41.750
And the row of zeros below,
which of course have no --
00:26:41.750 --> 00:26:43.860
that equation is
zero equals zero.
00:26:43.860 --> 00:26:46.080
That's satisfied.
00:26:46.080 --> 00:26:48.530
Here's my point.
00:26:48.530 --> 00:26:51.220
That when I do
back substitution,
00:26:51.220 --> 00:26:55.700
these numbers are
exactly what shows up --
00:26:55.700 --> 00:26:58.480
oh, their signs get switched.
00:26:58.480 --> 00:27:01.550
I was going to say those
numbers, two, minus two, zero,
00:27:01.550 --> 00:27:06.240
two, can I just circle the
-- this is the free part
00:27:06.240 --> 00:27:07.120
of the matrix.
00:27:07.120 --> 00:27:10.220
This is the identity part.
00:27:10.220 --> 00:27:14.710
This is the free part,
maybe I'll call it F.
00:27:14.710 --> 00:27:18.650
This, of course, I call I,
because it's the identity.
00:27:18.650 --> 00:27:25.020
The free part is a, I mean, I'm
just doing back substitution.
00:27:25.020 --> 00:27:29.340
And those free numbers will
show up, with a minus sign,
00:27:29.340 --> 00:27:32.040
because they pop onto the
other side of the equation --
00:27:32.040 --> 00:27:35.870
so I see minus two, zero,
and I see two, minus two.
00:27:39.670 --> 00:27:41.310
So that wasn't magic.
00:27:41.310 --> 00:27:43.540
It had to happen.
00:27:43.540 --> 00:27:48.240
Let me, show you
clearly why it happened.
00:27:48.240 --> 00:27:50.340
OK, so that's --
00:27:50.340 --> 00:27:53.290
this is what I'm
interested in here.
00:27:53.290 --> 00:27:59.410
And now let me, let me just,
like, do it, do it for --
00:27:59.410 --> 00:28:04.530
let's suppose we've,
we've got to --
00:28:07.750 --> 00:28:11.940
let's suppose we've got
this system already in,
00:28:11.940 --> 00:28:18.200
in rref form.
00:28:18.200 --> 00:28:22.580
So my matrix R is --
what does it look like?
00:28:22.580 --> 00:28:25.130
OK, and I'll --
00:28:25.130 --> 00:28:30.570
let me pretend that the
pivot columns come first
00:28:30.570 --> 00:28:33.220
and then whatever's
in the free columns.
00:28:33.220 --> 00:28:37.970
And there might be
some zero rows below.
00:28:37.970 --> 00:28:40.600
There's a typical --
00:28:40.600 --> 00:28:45.750
a pretty typical reduced
row echelon form.
00:28:49.450 --> 00:28:51.510
You see what's typical.
00:28:51.510 --> 00:28:55.430
It's got -- this is r by r.
00:28:55.430 --> 00:28:57.950
This is r pivot rows.
00:29:02.610 --> 00:29:06.050
This is r pivot columns.
00:29:09.780 --> 00:29:16.290
And here are n-r free columns.
00:29:16.290 --> 00:29:17.480
OK.
00:29:17.480 --> 00:29:21.650
Tell me what are the
special solutions?
00:29:21.650 --> 00:29:23.350
What are the --
00:29:23.350 --> 00:29:24.060
what's x?
00:29:24.060 --> 00:29:27.440
If I want to solve
R x equals zero --
00:29:27.440 --> 00:29:31.170
in fact, let me --
00:29:31.170 --> 00:29:36.110
I'm really going to, do
the whole -- since these --
00:29:36.110 --> 00:29:39.030
this is now block matrices,
I might as well do all
00:29:39.030 --> 00:29:41.300
of the special
solutions at once.
00:29:41.300 --> 00:29:44.740
So I want to solve
R x equals zero,
00:29:44.740 --> 00:29:49.660
and I'll have some
special solutions.
00:29:49.660 --> 00:29:52.830
Let me, actually --
00:29:52.830 --> 00:29:54.780
can I do them all at once?
00:29:54.780 --> 00:30:00.750
I'm going to create a
null space matrix, OK.
00:30:00.750 --> 00:30:01.790
A matrix.
00:30:04.970 --> 00:30:13.130
Its, its, its columns
are the special --
00:30:13.130 --> 00:30:15.085
the columns are the
special solutions.
00:30:19.080 --> 00:30:21.070
This is, I'm making
it sound harder,
00:30:21.070 --> 00:30:22.670
it's going to be totally easy.
00:30:22.670 --> 00:30:25.800
N will be this
null space matrix.
00:30:25.800 --> 00:30:31.070
I want R N to be
the zero matrix.
00:30:31.070 --> 00:30:33.830
These columns of N are
supposed to multipl-
00:30:33.830 --> 00:30:37.300
to get multiplied by R
and give zero columns.
00:30:37.300 --> 00:30:40.100
So what N will do the job?
00:30:40.100 --> 00:30:41.450
Let me put --
00:30:41.450 --> 00:30:45.290
I'm going to put the identity
in the free variable part
00:30:45.290 --> 00:30:54.020
and then minus F will show up
in the pivot variables, just
00:30:54.020 --> 00:30:55.980
the way it did in that example.
00:30:55.980 --> 00:30:58.970
There we had the identity and F.
00:30:58.970 --> 00:31:02.160
Here -- in the special solution.
00:31:02.160 --> 00:31:05.400
So these columns are --
there's the matrix of special
00:31:05.400 --> 00:31:06.430
solutions.
00:31:06.430 --> 00:31:09.460
And actually, there -- so
there's a Matlab command
00:31:09.460 --> 00:31:14.360
or a teaching code
command, NULL --
00:31:14.360 --> 00:31:19.460
N equal, so this is the --
00:31:19.460 --> 00:31:24.710
produces the null basis, the
null space matrix, NULL of A,
00:31:24.710 --> 00:31:26.230
and there it is.
00:31:30.450 --> 00:31:33.630
And how does that
command actually work?
00:31:33.630 --> 00:31:38.590
It uses Matlab to
compute R, then
00:31:38.590 --> 00:31:43.100
it picks out the pivot
variables, the free variables,
00:31:43.100 --> 00:31:47.980
puts, puts ones and zeros
in for the free variables,
00:31:47.980 --> 00:31:51.850
and copies out the
pivot variables.
00:31:51.850 --> 00:31:54.760
It, it does back substitution,
but back substitution
00:31:54.760 --> 00:31:57.180
for this system
is totally simple.
00:31:57.180 --> 00:32:00.070
What is this system?
00:32:00.070 --> 00:32:03.030
R x equals zero.
00:32:03.030 --> 00:32:11.620
So this is R is I F, and
x is the pivot variables
00:32:11.620 --> 00:32:18.240
and the free variables, and
it's supposed to give zero.
00:32:18.240 --> 00:32:20.240
So what does that mean?
00:32:20.240 --> 00:32:24.960
That means that the
pivot variables plus F
00:32:24.960 --> 00:32:28.640
times the free
variables give zero.
00:32:28.640 --> 00:32:31.950
So let me put F times the free
variables on the other side.
00:32:31.950 --> 00:32:37.950
I get minus F times
the free variables.
00:32:37.950 --> 00:32:43.530
There's my, equation,
as simple as it can be.
00:32:43.530 --> 00:32:45.840
That's what back
substitution comes
00:32:45.840 --> 00:32:49.280
to when I've reduced and
reduced and reduced this system
00:32:49.280 --> 00:32:52.090
to the, to the best form, OK.
00:32:52.090 --> 00:32:56.680
And, then if the
free variables, I
00:32:56.680 --> 00:33:00.070
make this special
choice of the identity,
00:33:00.070 --> 00:33:02.570
then the pivot variables are
00:33:02.570 --> 00:33:08.710
minus F. OK, can I
do, another example?
00:33:08.710 --> 00:33:10.180
Could you do another example?
00:33:10.180 --> 00:33:12.300
Can I -- let me just
take another matrix
00:33:12.300 --> 00:33:17.330
and, and let's go through
this algorithm once more, OK.
00:33:17.330 --> 00:33:19.180
Here we go.
00:33:19.180 --> 00:33:25.100
Here's a blackboard
for another matrix, OK.
00:33:25.100 --> 00:33:31.180
So I'll call the matrix A
again, but let me make it --
00:33:31.180 --> 00:33:33.680
yeah, how big shall
we make it this time?
00:33:36.480 --> 00:33:38.620
Why don't I do this?
00:33:38.620 --> 00:33:39.920
Just for the heck of it.
00:33:39.920 --> 00:33:44.860
Let me take the transpose of
this A and see what happens to
00:33:44.860 --> 00:33:45.670
that.
00:33:45.670 --> 00:33:55.820
Two four six eight and
three six eight ten.
00:34:00.250 --> 00:34:06.380
Before we do the calculations,
tell me what's coming?
00:34:06.380 --> 00:34:12.170
How many pivot variables
do you expect here?
00:34:12.170 --> 00:34:16.900
How many columns are
going to have pivots?
00:34:16.900 --> 00:34:22.060
How many -- we have three
columns in that matrix,
00:34:22.060 --> 00:34:25.739
but are we going to, are we
going to have three pivots?
00:34:25.739 --> 00:34:31.429
No, because this third columns
is the sum of the first two
00:34:31.429 --> 00:34:32.100
columns.
00:34:32.100 --> 00:34:38.350
I'm totally expecting, totally
expecting that these will be
00:34:38.350 --> 00:34:41.000
pivot columns --
00:34:41.000 --> 00:34:45.949
because they're independent,
but that this third guy,
00:34:45.949 --> 00:34:49.659
the third column, which is
dependent on the first two,
00:34:49.659 --> 00:34:52.219
is going to be a free column.
00:34:52.219 --> 00:34:54.889
Elimination better
discover that.
00:34:54.889 --> 00:34:58.270
And elimination will
also straighten out
00:34:58.270 --> 00:35:05.170
the rows, dependent rows
and some independent rows.
00:35:05.170 --> 00:35:09.510
What's the, what's the row
reduced echelon form for this?
00:35:09.510 --> 00:35:11.970
Let's just do it, OK.
00:35:11.970 --> 00:35:16.710
So, so that's the first pivot.
00:35:16.710 --> 00:35:20.360
Two times that away from
that gives me a row of zeros.
00:35:20.360 --> 00:35:24.970
Two times that away from
that gives me a zero two two.
00:35:24.970 --> 00:35:28.865
And two times that away from
that gives me a zero four four.
00:35:32.300 --> 00:35:35.750
OK, first column is straight.
00:35:35.750 --> 00:35:37.780
First variable is
a pivot variable.
00:35:37.780 --> 00:35:39.010
No problem.
00:35:39.010 --> 00:35:40.370
On to the second column.
00:35:40.370 --> 00:35:43.880
I look at the second
pivot, it's a zero.
00:35:43.880 --> 00:35:45.430
I look below it.
00:35:45.430 --> 00:35:46.740
There's a two.
00:35:46.740 --> 00:35:47.890
OK, I do a row exchange.
00:35:50.530 --> 00:35:53.630
So this zero is now there.
00:35:53.630 --> 00:35:58.710
I now have a perfectly
good pivot, and I use it.
00:35:58.710 --> 00:36:03.030
OK, and I subtract two of
that row away from this row.
00:36:03.030 --> 00:36:05.630
All right if I do it like that?
00:36:05.630 --> 00:36:08.310
I've got to the form U now.
00:36:08.310 --> 00:36:10.460
This was my A.
00:36:10.460 --> 00:36:17.260
Now there's my U.
I can see now --
00:36:17.260 --> 00:36:18.630
I have to stop, right?
00:36:18.630 --> 00:36:20.340
I would go on to
the third column.
00:36:20.340 --> 00:36:22.650
I should have tried.
00:36:22.650 --> 00:36:24.120
I quit, but without trying.
00:36:24.120 --> 00:36:25.440
I shouldn't have done that.
00:36:25.440 --> 00:36:28.810
On to the third column,
look at the pivot position.
00:36:28.810 --> 00:36:30.100
It's got a zero in it.
00:36:30.100 --> 00:36:32.300
Look below, all zeros.
00:36:32.300 --> 00:36:35.660
Now I'm entitled to stop, OK.
00:36:35.660 --> 00:36:37.445
So the rank is two again.
00:36:45.860 --> 00:36:48.440
What about the null space?
00:36:48.440 --> 00:36:52.450
How many special solutions
are there this time?
00:36:52.450 --> 00:36:56.990
How many special
solutions for this matrix?
00:36:56.990 --> 00:36:59.600
I've got -- and which are the
free variables and which are
00:36:59.600 --> 00:37:01.400
the pivot variables and so on?
00:37:01.400 --> 00:37:04.610
Pivot columns, I've
got two pivot columns,
00:37:04.610 --> 00:37:06.890
and that's no accident.
00:37:06.890 --> 00:37:11.200
The number of pivot columns
for a matrix A, that's
00:37:11.200 --> 00:37:16.120
a great fact, that the
number of pivot columns for A
00:37:16.120 --> 00:37:19.250
and A transpose are the same.
00:37:19.250 --> 00:37:21.680
And then I have a free column.
00:37:21.680 --> 00:37:24.240
There's a free column.
00:37:24.240 --> 00:37:30.330
One free column, because the
count is three minus two.
00:37:30.330 --> 00:37:33.990
Three minus two gives
me one free column.
00:37:37.760 --> 00:37:47.410
OK, so now let me solve,
what's in the null space.
00:37:47.410 --> 00:37:49.390
OK, so how do I --
00:37:49.390 --> 00:37:50.530
let's see.
00:37:50.530 --> 00:37:52.800
These vectors have length three.
00:37:52.800 --> 00:37:54.480
They only have three components.
00:37:54.480 --> 00:37:58.200
I'm making too much space
for the, to write x.
00:37:58.200 --> 00:38:03.320
x has just got three
components, and what are they?
00:38:03.320 --> 00:38:06.340
I'm looking for the null space.
00:38:09.660 --> 00:38:12.430
OK, so how do I start?
00:38:12.430 --> 00:38:17.710
I give the free variable
some convenient value.
00:38:17.710 --> 00:38:20.290
And what's that?
00:38:20.290 --> 00:38:23.000
I set it to one.
00:38:23.000 --> 00:38:24.970
I set the free variable to one.
00:38:24.970 --> 00:38:28.520
If I set the free
variable to zero and solve
00:38:28.520 --> 00:38:33.300
for the pivot variables, I'll
get all zeros: no progress.
00:38:33.300 --> 00:38:36.430
But by setting the
free variable to one --
00:38:36.430 --> 00:38:39.680
you see w- my two
equations now are --
00:38:39.680 --> 00:38:45.230
my equations are x1+
2x2+ 3 x3 is zero,
00:38:45.230 --> 00:38:47.200
that's my first equation.
00:38:47.200 --> 00:38:51.130
And my second equation is
now 2x2+2x3 equals zero.
00:38:51.130 --> 00:38:56.110
And, OK.
00:38:56.110 --> 00:39:02.400
So if x3 is one,
then x2 is minus one.
00:39:02.400 --> 00:39:07.430
And if x3 is one and x2 is
minus one, then maybe x1
00:39:07.430 --> 00:39:08.510
is minus one.
00:39:12.080 --> 00:39:14.360
And actually I go
back to check now.
00:39:14.360 --> 00:39:17.410
I don't, like --
00:39:17.410 --> 00:39:19.410
I did a quick
calculation mentally.
00:39:19.410 --> 00:39:21.570
Can I mentally do a quick check?
00:39:21.570 --> 00:39:24.180
That matrix, that
solution x says
00:39:24.180 --> 00:39:28.710
that minus this column minus
this column plus this one
00:39:28.710 --> 00:39:31.560
is the zero column.
00:39:31.560 --> 00:39:32.860
And it is.
00:39:32.860 --> 00:39:35.510
Minus that minus that
plus that is zero.
00:39:35.510 --> 00:39:37.720
So that's in the null space.
00:39:37.720 --> 00:39:40.790
And now you can tell me what
else is in the null space.
00:39:40.790 --> 00:39:44.080
What's, what's the
whole null space now?
00:39:44.080 --> 00:39:46.570
I multiply by C, right.
00:39:46.570 --> 00:39:51.720
The whole null space is a line.
00:39:51.720 --> 00:39:52.970
So that's the description.
00:39:52.970 --> 00:39:57.520
You know, if I ask you on a
homework or a quiz or the final
00:39:57.520 --> 00:40:01.630
what -- give me, describe,
tell me the null space,
00:40:01.630 --> 00:40:04.410
find the null space
of this matrix,
00:40:04.410 --> 00:40:07.340
you can take those steps.
00:40:07.340 --> 00:40:10.790
And that's the answer
I'm looking for.
00:40:10.790 --> 00:40:15.190
And I'm looking for that C
too, because that's telling me
00:40:15.190 --> 00:40:18.360
that you're remembering that
it's a whole space and not
00:40:18.360 --> 00:40:20.640
just one vector.
00:40:20.640 --> 00:40:23.960
Later I will ask you for a
basis for the null space.
00:40:23.960 --> 00:40:26.850
Then I just want this vector.
00:40:26.850 --> 00:40:28.840
But if I ask for the
whole null space,
00:40:28.840 --> 00:40:31.510
then there's the whole
line through that vector.
00:40:31.510 --> 00:40:36.340
OK, now one more natural thing
to do with this example, right,
00:40:36.340 --> 00:40:43.300
is keep going to the
reduced matrix, R.
00:40:43.300 --> 00:40:46.180
So can I push onwards to R?
00:40:46.180 --> 00:40:49.340
That should be quick,
but let's just practice.
00:40:49.340 --> 00:40:54.000
Let me keep going to R.
OK, so what do I do here?
00:40:54.000 --> 00:40:55.970
I subtract --
00:40:55.970 --> 00:40:57.830
I clear out above
the pivot, so I
00:40:57.830 --> 00:41:02.870
subtract that from that, that's
one zero one is left, right?
00:41:02.870 --> 00:41:04.740
When I subtracted
this row from this
00:41:04.740 --> 00:41:07.390
it produced a zero
above this pivot.
00:41:07.390 --> 00:41:12.070
And now I want that
pivot to be a one.
00:41:12.070 --> 00:41:18.160
So for the R matrix, I'll
divide this equation by two,
00:41:18.160 --> 00:41:23.270
and of course these zero, zeros
are great, they don't change.
00:41:23.270 --> 00:41:24.700
There's R.
00:41:24.700 --> 00:41:27.370
That's R.
00:41:27.370 --> 00:41:28.430
You see what R is?
00:41:28.430 --> 00:41:32.100
You see the identity
matrix sitting up here?
00:41:32.100 --> 00:41:36.370
You see the free part
F, the F part here?
00:41:36.370 --> 00:41:38.370
And you see the zeros below.
00:41:38.370 --> 00:41:42.460
This is I F zero zero.
00:41:42.460 --> 00:41:45.180
And what's the x?
00:41:45.180 --> 00:41:48.660
The x has the identity --
00:41:48.660 --> 00:41:51.460
well, it's only a
single number one,
00:41:51.460 --> 00:41:56.090
but it's the identity matrix
in the free, in the free part.
00:41:56.090 --> 00:42:00.770
And what does it have
in the pivot variables?
00:42:00.770 --> 00:42:03.620
What did back substitution give?
00:42:03.620 --> 00:42:06.910
It gave minus these guys.
00:42:06.910 --> 00:42:11.160
You see that what this
is is any multiple of --
00:42:11.160 --> 00:42:14.360
this is the identity there,
and this is minus F here.
00:42:19.190 --> 00:42:24.710
This is our null space
matrix N for this.
00:42:24.710 --> 00:42:28.490
Our, our null space matrix
is the guy whose columns
00:42:28.490 --> 00:42:30.680
are the special solutions.
00:42:30.680 --> 00:42:33.570
So their free variables
have the special values
00:42:33.570 --> 00:42:39.510
one and, pivot
variables have minus F.
00:42:39.510 --> 00:42:42.210
So do you see, though,
how the minus F just
00:42:42.210 --> 00:42:45.540
automatically shows up
in the special solutions.
00:42:49.390 --> 00:42:50.389
That's it really.
00:42:50.389 --> 00:42:51.930
I don't think there's
anything more I
00:42:51.930 --> 00:42:55.980
can say about A x equals zero.
00:42:55.980 --> 00:42:59.640
There's more I can
say about A x equal b,
00:42:59.640 --> 00:43:03.080
but that'll be on Friday.
00:43:03.080 --> 00:43:05.370
OK, so that's, that's
the null space.
00:43:05.370 --> 00:43:06.920
Thanks.