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I've been multiplying
matrices already,
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but certainly time for
me to discuss the rules
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for matrix multiplication.
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And the interesting part is
the many ways you can do it,
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and they all give
the same answer.
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And they're all important.
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So matrix multiplication,
and then, come inverses.
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So we mentioned the
inverse of a matrix.
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That's a big deal.
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Lots to do about inverses
and how to find them.
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Okay, so I'll begin with how
to multiply two matrices.
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First way, okay, so suppose
I have a matrix A multiplying
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a matrix B and --
giving me a result --
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well, I could call it C.
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A times B.
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Okay.
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So, let me just review
the rule for this entry.
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That's the entry in
row i and column j.
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So that's the i j entry.
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Right there is C i j.
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We always write the row number
and then the column number.
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So I might --
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I might -- maybe I take it C
3 4, just to make it specific.
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So instead of i j,
let me use numbers.
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C 3 4.
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So where does that come
from, the three four entry?
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It comes from row three, here,
row three and column four,
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as you know.
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Column four.
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And can I just write
down, or can we
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write down the formula for it?
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If we look at the whole
row and the whole column,
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the quick way for me to
say it is row three of A --
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I could use a dot
for dot product.
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I won't often use
that, actually.
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Dot column four of B.
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But this gives us a
chance to just, like,
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use a little matrix notation.
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What are the entries?
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What's this first
entry in row three?
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That number that's
sitting right there is...
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A, so it's got two
indices and what are they?
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3 1.
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So there's an a 3 1 there.
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Now what's the first guy
at the top of column four?
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So what's sitting up there?
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B 1 4, right.
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So that this dot product
starts with A 3 1 times B 1 4.
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And then what's the next -- so
this is like I'm accumulating
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this sum, then comes the next
guy, A 3 2, second column,
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times B 2 4, second row.
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So it's b A 3 2,
B 2 4 and so on.
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Just practice with indices.
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Oh, let me even practice
with a summation formula.
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So this is --
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most of the course,
I use whole vectors.
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I very seldom, get
down to the details
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of these particular entries,
but here we'd better do it.
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So it's some kind
of a sum, right?
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Of things in row three,
column K shall I say?
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Times things in
row K, column four.
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Do you see that that's
what we're seeing here?
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This is K is one, here
K is two, on along --
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so the sum goes all the
way along the row and down
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the column, say, one to N.
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So that's what the C three
four entry looks like.
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A sum of a three K b K four.
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Just takes a little
practice to do that.
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Okay.
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And -- well, maybe
I should say --
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when are we allowed to
multiply these matrices?
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What are the shapes
of these things?
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The shapes are --
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if we allow them to be not
necessarily square matrices.
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If they're square,
they've got to be the same
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size.
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If they're rectangular,
they're not the same size.
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If they're rectangular,
this might be -- well,
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I always think of A as m by n.
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m rows, n columns.
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So that sum goes to n.
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Now what's the point -- how
many rows does B have to have?
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n.
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The number of rows in
B, the number of guys
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that we meet coming down has
to match the number of ones
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across.
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So B will have to
be n by something.
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Whatever.
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P. So the number of columns here
has to match the number of rows
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there, and then
what's the result?
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What's the shape of the result?
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What's the shape
of C, the output?
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Well, it's got these same
m rows -- it's got m rows.
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And how many columns?
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P. m by P. Okay.
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So there are m times P little
numbers in there, entries,
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and each one, looks like that.
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Okay.
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So that's the standard rule.
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That's the way people think
of multiplying matrices.
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I do it too.
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But I want to talk
about other ways
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to look at that
same calculation,
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looking at whole
columns and whole rows.
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Okay.
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So can I do A B C again?
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A B equaling C again?
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But now, tell me about...
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I'll put it up here.
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So here goes A, again,
times B producing C.
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And again, this is m by n.
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This is n by P and
this is m by P. Okay.
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Now I want to look
at whole columns.
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I want to look at
the columns of --
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here's the second way
to multiply matrices.
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Because I'm going to build
on what I know already.
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How do I multiply a
matrix by a column?
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I know how to multiply
this matrix by that column.
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Shall I call that column one?
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That tells me column
one of the answer.
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The matrix times the first
column is that first column.
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Because none of
this stuff entered
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that part of the answer.
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The matrix times
the second column
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is the second column
of the answer.
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Do you see what I'm saying?
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That I could think of
multiplying a matrix
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by a vector, which I
already knew how to do,
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and I can think of just P
columns sitting side by side,
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just like resting
next to each other.
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And I multiply A times
each one of those.
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And I get the P
columns of the answer.
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Do you see this as --
this is quite nice,
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to be able to think, okay,
matrix multiplication works
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so that I can just think
of having several columns,
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multiplying by A and getting
the columns of the answer.
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So, like, here's column one
shall I call that column one?
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And what's going in there
is A times column one.
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Okay.
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So that's the picture
a column at a time.
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So what does that tell me?
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What does that tell me
about these columns?
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These columns of C
are combinations,
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because we've seen that
before, of columns of A.
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Every one of these
comes from A times this,
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and A times a vector
is a combination
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of the columns of A.
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And it makes sense, because
the columns of A have length m
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and the columns of
C have length m.
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And every column of
C is some combination
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of the columns of A.
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And it's these
numbers in here that
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tell me what combination it is.
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Do you see that?
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That in that answer, C,
I'm seeing stuff that's
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combinations of these columns.
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Now, suppose I look at it
-- that's two ways now.
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The third way is
look at it by rows.
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So now let me change to rows.
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Okay.
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So now I can think
of a row of A --
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a row of A multiplying all these
rows here and producing a row
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of the product.
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So this row takes a
combination of these rows
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and that's the answer.
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So these rows of C are
combinations of what?
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Tell me how to finish that.
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The rows of C, when I have a
matrix B, it's got its rows
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and I multiply by A,
and what does that do?
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It mixes the rows up.
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It creates combinations
of the rows of B, thanks.
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Rows of B.
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That's what I wanted to
see, that this answer --
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I can see where the
pieces are coming from.
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The rows in the answer are
coming as combinations of these
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rows.
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The columns in the answer are
coming as combinations of those
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columns.
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And so that's three ways.
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Now you can say, okay,
what's the fourth way?
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The fourth way --
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so that's -- now we've
got, like, the regular way,
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the column way,
the row way and --
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what's left?
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The one that I can --
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well, one way is
columns times rows.
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What happens if I multiply --
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So this was row times
column, it gave a number.
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Okay.
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00:12:21,980 --> 00:12:25,670
Now I want to ask you
about column times row.
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If I multiply a column
of A times a row of B,
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what shape I ending up with?
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So if I take a
column times a row,
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that's definitely different from
taking a row times a column.
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So a column of A was -- what's
the shape of a column of A?
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m by one.
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A column of A is a column.
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00:12:59,430 --> 00:13:03,190
It's got m entries
and one column.
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And what's a row of B?
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It's got one row and P columns.
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So what's the shape -- what do
I get if I multiply a column
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00:13:12,870 --> 00:13:13,590
by a row?
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I get a big matrix.
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I get a full-sized matrix.
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00:13:20,540 --> 00:13:25,790
If I multiply a
column by a row --
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00:13:25,790 --> 00:13:28,090
should we just do one?
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00:13:28,090 --> 00:13:34,470
Let me take the column two three
four times the row one six.
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00:13:38,510 --> 00:13:40,825
That product there --
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00:13:40,825 --> 00:13:42,950
I mean, when I'm just
following the rules of matrix
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00:13:42,950 --> 00:13:46,490
multiplication, those rules
are just looking like --
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00:13:46,490 --> 00:13:51,570
kind of petite, kind of
small, because the rows here
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00:13:51,570 --> 00:13:53,830
are so short and the
columns there are so short,
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00:13:53,830 --> 00:13:56,610
but they're the same
length, one entry.
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00:13:56,610 --> 00:13:57,530
So what's the answer?
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00:14:00,270 --> 00:14:03,770
What's the answer if I do two
three four times one six, just
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00:14:03,770 --> 00:14:05,320
for practice?
217
00:14:05,320 --> 00:14:09,180
Well, what's the first
row of the answer?
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00:14:09,180 --> 00:14:11,030
Two twelve.
219
00:14:11,030 --> 00:14:16,420
And the second row of the
answer is three eighteen.
220
00:14:16,420 --> 00:14:24,220
And the third row of the
answer is four twenty four.
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00:14:24,220 --> 00:14:29,400
That's a very special
matrix, there.
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00:14:29,400 --> 00:14:30,840
Very special matrix.
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00:14:30,840 --> 00:14:32,830
What can you tell me
about its columns,
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00:14:32,830 --> 00:14:34,220
the columns of that matrix?
225
00:14:37,640 --> 00:14:41,410
They're multiples
of this guy, right?
226
00:14:41,410 --> 00:14:42,690
They're multiples of that one.
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00:14:42,690 --> 00:14:44,340
Which follows our rule.
228
00:14:44,340 --> 00:14:47,540
We said that the columns of
the answer were combinations,
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00:14:47,540 --> 00:14:50,300
but there's only -- to take
a combination of one guy,
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it's just a multiple.
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00:14:52,300 --> 00:14:54,000
The rows of the
answer, what can you
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00:14:54,000 --> 00:14:55,530
tell me about those three rows?
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00:14:58,060 --> 00:15:01,480
They're all multiples
of this row.
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00:15:01,480 --> 00:15:04,660
They're all multiples of
one six, as we expected.
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But I'm getting a
full-sized matrix.
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00:15:06,970 --> 00:15:15,800
And now, just to complete
this thought, if I have --
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00:15:15,800 --> 00:15:17,280
let me write down
the fourth way.
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00:15:21,110 --> 00:15:35,630
A B is a sum of columns
of A times rows of B.
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00:15:35,630 --> 00:15:40,680
So that, for example, if my
matrix was two three four
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00:15:40,680 --> 00:15:47,450
and then had another column,
say, seven eight nine,
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00:15:47,450 --> 00:15:53,010
and my matrix here has -- say,
started with one six and then
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had another column
like zero zero, then --
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00:16:00,890 --> 00:16:04,630
here's the fourth way, okay?
244
00:16:04,630 --> 00:16:07,690
I've got two columns there,
I've got two rows there.
245
00:16:07,690 --> 00:16:10,920
So the beautiful rule is --
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00:16:10,920 --> 00:16:13,080
see, the whole thing
by columns and rows
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00:16:13,080 --> 00:16:19,730
is that I can take the first
column times the first row
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00:16:19,730 --> 00:16:25,100
and add the second column
times the second row.
249
00:16:32,690 --> 00:16:34,790
So that's the fourth way --
250
00:16:34,790 --> 00:16:38,760
that I can take
columns times rows,
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00:16:38,760 --> 00:16:41,800
first column times first row,
second column times second
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00:16:41,800 --> 00:16:42,960
row and add.
253
00:16:42,960 --> 00:16:44,110
Actually, what will I get?
254
00:16:44,110 --> 00:16:46,765
What will the answer be for
that matrix multiplication?
255
00:16:49,730 --> 00:16:52,130
Well, this one it's just
going to give us zero,
256
00:16:52,130 --> 00:16:56,070
so in fact I'm back to
this -- that's the answer,
257
00:16:56,070 --> 00:16:59,310
for that matrix multiplication.
258
00:16:59,310 --> 00:17:05,569
I'm happy to put up here
these facts about matrix
259
00:17:05,569 --> 00:17:10,040
multiplication, because it
gives me a chance to write down
260
00:17:10,040 --> 00:17:12,099
special matrices like this.
261
00:17:12,099 --> 00:17:15,050
This is a special matrix.
262
00:17:15,050 --> 00:17:18,569
All those rows lie
on the same line.
263
00:17:18,569 --> 00:17:21,900
All those rows lie on
the line through one six.
264
00:17:21,900 --> 00:17:25,240
If I draw a picture of
all these row vectors,
265
00:17:25,240 --> 00:17:28,119
they're all the same direction.
266
00:17:28,119 --> 00:17:30,730
If I draw a picture of
these two column vectors,
267
00:17:30,730 --> 00:17:33,900
they're in the same direction.
268
00:17:33,900 --> 00:17:37,430
Later, I would
use this language.
269
00:17:37,430 --> 00:17:39,310
Not too much later, either.
270
00:17:39,310 --> 00:17:42,400
I would say the
row space, which is
271
00:17:42,400 --> 00:17:44,380
like all the
combinations of the rows,
272
00:17:44,380 --> 00:17:47,060
is just a line for this matrix.
273
00:17:47,060 --> 00:17:51,240
The row space is the line
through the vector one six.
274
00:17:51,240 --> 00:17:54,390
All the rows lie on that line.
275
00:17:54,390 --> 00:17:57,710
And the column space
is also a line.
276
00:17:57,710 --> 00:18:01,510
All the columns lie on the
line through the vector two
277
00:18:01,510 --> 00:18:02,950
three four.
278
00:18:02,950 --> 00:18:07,340
So this is like a
really minimal matrix.
279
00:18:07,340 --> 00:18:10,940
And it's because of these ones.
280
00:18:10,940 --> 00:18:11,440
Okay.
281
00:18:11,440 --> 00:18:18,580
So that's a third way.
282
00:18:18,580 --> 00:18:23,160
Now I want to say one more thing
about matrix multiplication
283
00:18:23,160 --> 00:18:26,100
while we're on the subject.
284
00:18:26,100 --> 00:18:28,060
And it's this.
285
00:18:28,060 --> 00:18:29,740
You could also multiply --
286
00:18:29,740 --> 00:18:34,390
You could also cut
the matrix into blocks
287
00:18:34,390 --> 00:18:37,160
and do the
multiplication by blocks.
288
00:18:37,160 --> 00:18:46,690
Yet that's actually so, useful
that I want to mention it.
289
00:18:46,690 --> 00:18:47,745
Block multiplication.
290
00:18:50,540 --> 00:18:54,500
So I could take my matrix A
and I could chop it up, like,
291
00:18:54,500 --> 00:18:58,460
maybe just for simplicity,
let me chop it into two --
292
00:18:58,460 --> 00:18:59,770
into four square blocks.
293
00:18:59,770 --> 00:19:00,980
Suppose it's square.
294
00:19:00,980 --> 00:19:03,990
Let's just take a nice case.
295
00:19:03,990 --> 00:19:07,800
And B, suppose it's
square also, same size.
296
00:19:11,340 --> 00:19:13,460
So these sizes don't
have to be the same.
297
00:19:13,460 --> 00:19:16,540
What they have to do
is match properly.
298
00:19:16,540 --> 00:19:18,590
Here they certainly will match.
299
00:19:18,590 --> 00:19:22,340
So here's the rule for
block multiplication,
300
00:19:22,340 --> 00:19:28,880
that if this has
blocks like, A --
301
00:19:28,880 --> 00:19:34,220
so maybe A1, A2, A3,
A4 are the blocks here,
302
00:19:34,220 --> 00:19:38,210
and these blocks
are B1, B2,3 and B4?
303
00:19:38,210 --> 00:19:44,770
Then the answer
I can find block.
304
00:19:44,770 --> 00:19:46,500
And if you tell me
what's in that block,
305
00:19:46,500 --> 00:19:49,560
then I'm going to be quiet
about matrix multiplication
306
00:19:49,560 --> 00:19:51,870
for the rest of the day.
307
00:19:51,870 --> 00:19:54,920
What goes into that block?
308
00:19:54,920 --> 00:19:57,870
You see, these might be
-- this matrix might be --
309
00:19:57,870 --> 00:20:02,400
these matrices might be, like,
twenty by twenty with blocks
310
00:20:02,400 --> 00:20:06,030
that are ten by ten, to take the
easy case where all the blocks
311
00:20:06,030 --> 00:20:08,940
are the same shape.
312
00:20:08,940 --> 00:20:13,110
And the point is that I could
multiply those by blocks.
313
00:20:13,110 --> 00:20:15,940
And what goes in here?
314
00:20:15,940 --> 00:20:20,920
What's that block in the answer?
315
00:20:20,920 --> 00:20:25,710
A1 B1, that's a
matrix times a matrix,
316
00:20:25,710 --> 00:20:28,270
it's the right size, ten by ten.
317
00:20:28,270 --> 00:20:30,610
Any more?
318
00:20:30,610 --> 00:20:36,760
Plus, what else goes in there?
319
00:20:36,760 --> 00:20:38,160
A2 B3, right?
320
00:20:38,160 --> 00:20:41,490
It's just like block
rows times block columns.
321
00:20:45,610 --> 00:20:49,070
Nobody, I think, not even
Gauss could see instantly
322
00:20:49,070 --> 00:20:51,040
that it works.
323
00:20:51,040 --> 00:20:55,130
But somehow, if we check
it through, all five ways
324
00:20:55,130 --> 00:20:58,020
we're doing the same
multiplications.
325
00:20:58,020 --> 00:21:02,430
So this familiar
multiplication is
326
00:21:02,430 --> 00:21:04,270
what we're really
doing when we do it
327
00:21:04,270 --> 00:21:10,240
by columns, by rows by columns
times rows and by blocks.
328
00:21:10,240 --> 00:21:10,740
Okay.
329
00:21:10,740 --> 00:21:13,680
I just have to, like,
get the rules straight
330
00:21:13,680 --> 00:21:16,991
for matrix multiplication.
331
00:21:16,991 --> 00:21:17,490
Okay.
332
00:21:22,740 --> 00:21:24,490
All right, I'm ready
for the second topic,
333
00:21:24,490 --> 00:21:27,590
which is inverses.
334
00:21:27,590 --> 00:21:28,479
Okay.
335
00:21:28,479 --> 00:21:29,270
Ready for inverses.
336
00:21:34,140 --> 00:21:39,040
And let me do it for
square matrices first.
337
00:21:44,270 --> 00:21:44,770
Okay.
338
00:21:44,770 --> 00:21:52,270
So I've got a square matrix A.
339
00:21:52,270 --> 00:21:55,370
And it may or may not
have an inverse, right?
340
00:21:55,370 --> 00:21:57,260
Not all matrices have inverses.
341
00:21:57,260 --> 00:22:01,770
In fact, that's the most
important question you can ask
342
00:22:01,770 --> 00:22:06,260
about the matrix, is if it's
-- if you know it's square,
343
00:22:06,260 --> 00:22:08,540
is it invertible or not?
344
00:22:08,540 --> 00:22:12,500
If it is invertible, then
there is some other matrix,
345
00:22:12,500 --> 00:22:15,680
shall I call it A inverse?
346
00:22:15,680 --> 00:22:21,600
And what's the -- if
A inverse exists --
347
00:22:21,600 --> 00:22:24,270
there's a big "if" here.
348
00:22:24,270 --> 00:22:34,120
If this matrix exists, and it'll
be really central to figure out
349
00:22:34,120 --> 00:22:35,940
when does it exist?
350
00:22:35,940 --> 00:22:40,350
And then if it does exist,
how would you find it?
351
00:22:40,350 --> 00:22:45,710
But what's the equation
here that I haven't --
352
00:22:45,710 --> 00:22:47,900
that I have to finish now?
353
00:22:47,900 --> 00:22:53,280
This matrix, if it exists
multiplies A and produces,
354
00:22:53,280 --> 00:22:54,755
I think, the identity.
355
00:23:10,840 --> 00:23:12,680
But a real --
356
00:23:12,680 --> 00:23:17,600
an inverse for a square matrix
could be on the right as well
357
00:23:17,600 --> 00:23:19,240
--
358
00:23:19,240 --> 00:23:25,760
this is true, too, that it's --
359
00:23:25,760 --> 00:23:28,280
if I have a -- yeah in
fact, this is not --
360
00:23:28,280 --> 00:23:31,570
this is probably the --
361
00:23:31,570 --> 00:23:38,400
this is something that's not
easy to prove, but it works.
362
00:23:38,400 --> 00:23:40,240
That a left --
363
00:23:40,240 --> 00:23:43,670
square matrices, a left
inverse is also a right
364
00:23:43,670 --> 00:23:44,570
inverse.
365
00:23:44,570 --> 00:23:49,090
If I can find a matrix on the
left that gets the identity,
366
00:23:49,090 --> 00:23:50,990
then also that
matrix on the right
367
00:23:50,990 --> 00:23:53,630
will produce that identity.
368
00:23:53,630 --> 00:23:57,310
For rectangular matrices,
we'll see a left inverse
369
00:23:57,310 --> 00:23:58,950
that isn't a right inverse.
370
00:23:58,950 --> 00:24:01,660
In fact, the shapes
wouldn't allow it.
371
00:24:01,660 --> 00:24:03,700
But for square
matrices, the shapes
372
00:24:03,700 --> 00:24:09,820
allow it and it happens,
if A has an inverse.
373
00:24:09,820 --> 00:24:14,011
Okay, so give me some cases --
374
00:24:14,011 --> 00:24:14,510
let's see.
375
00:24:14,510 --> 00:24:17,090
I hate to be negative
here, but let's talk
376
00:24:17,090 --> 00:24:20,070
about the case with no inverse.
377
00:24:20,070 --> 00:24:31,210
So -- these matrices are called
invertible or non-singular --
378
00:24:36,640 --> 00:24:38,960
those are the good ones.
379
00:24:38,960 --> 00:24:41,510
And we want to be able
to identify how --
380
00:24:41,510 --> 00:24:44,180
if we're given a matrix,
has it got an inverse?
381
00:24:44,180 --> 00:24:46,950
Can I talk about
the singular case?
382
00:24:52,210 --> 00:24:52,940
No inverse.
383
00:24:57,420 --> 00:24:58,950
All right.
384
00:24:58,950 --> 00:25:02,720
Best to start with an example.
385
00:25:02,720 --> 00:25:06,880
Tell me an example -- let's
get an example up here.
386
00:25:06,880 --> 00:25:09,030
Let's make it two by two --
387
00:25:09,030 --> 00:25:13,540
of a matrix that has
not got an inverse.
388
00:25:13,540 --> 00:25:16,160
And let's see why.
389
00:25:16,160 --> 00:25:19,230
Let me write one up.
390
00:25:19,230 --> 00:25:20,950
No inverse.
391
00:25:20,950 --> 00:25:22,620
Let's see why.
392
00:25:22,620 --> 00:25:30,780
Let me write up --
one three two six.
393
00:25:35,180 --> 00:25:37,820
Why does that matrix
have no inverse?
394
00:25:40,950 --> 00:25:43,245
You could answer
that various ways.
395
00:25:45,760 --> 00:25:48,390
Give me one reason.
396
00:25:48,390 --> 00:25:51,620
Well, you could -- if you
know about determinants,
397
00:25:51,620 --> 00:25:57,060
which you're not supposed to,
you could take its determinant
398
00:25:57,060 --> 00:25:58,740
and you would get --
399
00:25:58,740 --> 00:26:00,330
Zero.
400
00:26:00,330 --> 00:26:03,090
Okay.
401
00:26:03,090 --> 00:26:04,120
Now -- all right.
402
00:26:07,640 --> 00:26:10,330
Let me ask you other reasons.
403
00:26:10,330 --> 00:26:13,120
I mean, as for other
reasons that that matrix
404
00:26:13,120 --> 00:26:15,810
isn't invertible.
405
00:26:15,810 --> 00:26:18,765
Here, I could use
what I'm saying here.
406
00:26:24,550 --> 00:26:27,800
Suppose A times other
matrix gave the identity.
407
00:26:32,060 --> 00:26:34,750
Why is that not possible?
408
00:26:34,750 --> 00:26:39,010
Because -- oh, yeah --
409
00:26:39,010 --> 00:26:41,220
I'm thinking about columns here.
410
00:26:41,220 --> 00:26:46,200
If I multiply this matrix A by
some other matrix, then the --
411
00:26:46,200 --> 00:26:50,240
the result -- what can you
tell me about the columns?
412
00:26:50,240 --> 00:26:55,730
They're all multiples
of those columns, right?
413
00:26:55,730 --> 00:26:59,340
If I multiply A by
another matrix that --
414
00:26:59,340 --> 00:27:04,130
the product has columns that
come from those columns.
415
00:27:04,130 --> 00:27:06,480
So can I get the
identity matrix?
416
00:27:06,480 --> 00:27:08,120
No way.
417
00:27:08,120 --> 00:27:11,670
The columns of the identity
matrix, like one zero --
418
00:27:11,670 --> 00:27:14,740
it's not a combination
of those columns,
419
00:27:14,740 --> 00:27:16,520
because those two
columns lie on the --
420
00:27:16,520 --> 00:27:19,140
both lie on the same line.
421
00:27:19,140 --> 00:27:22,140
Every combination is just
going to be on that line
422
00:27:22,140 --> 00:27:24,480
and I can't get one zero.
423
00:27:24,480 --> 00:27:33,530
So, do you see that sort of
column picture of the matrix
424
00:27:33,530 --> 00:27:34,750
not being invertible.
425
00:27:34,750 --> 00:27:37,580
In fact, here's another reason.
426
00:27:37,580 --> 00:27:40,720
This is even a more
important reason.
427
00:27:40,720 --> 00:27:42,260
Well, how can I
say more important?
428
00:27:42,260 --> 00:27:44,810
All those are important.
429
00:27:44,810 --> 00:27:47,910
This is another way to see it.
430
00:27:47,910 --> 00:27:51,410
A matrix has no inverse --
431
00:27:51,410 --> 00:27:55,710
yeah -- here -- now
this is important.
432
00:27:55,710 --> 00:27:59,180
A matrix has no -- a square
matrix won't have an inverse
433
00:27:59,180 --> 00:28:07,980
if there's no inverse
because I can solve --
434
00:28:07,980 --> 00:28:20,810
I can find an X of -- a
vector X with A times --
435
00:28:20,810 --> 00:28:23,140
this A times X giving zero.
436
00:28:27,010 --> 00:28:31,310
This is the reason I like best.
437
00:28:31,310 --> 00:28:33,170
That matrix won't
have an inverse.
438
00:28:33,170 --> 00:28:40,490
Can you -- well, let
me change I to U.
439
00:28:40,490 --> 00:28:46,780
So tell me a vector X that,
solves A X equals zero.
440
00:28:46,780 --> 00:28:49,610
I mean, this is, like,
the key equation.
441
00:28:49,610 --> 00:28:51,310
In mathematics, all
the key equations
442
00:28:51,310 --> 00:28:53,490
have zero on the
right-hand side.
443
00:28:53,490 --> 00:28:55,490
So what's the X?
444
00:28:55,490 --> 00:28:58,290
Tell me an X here --
445
00:28:58,290 --> 00:29:01,360
so now I'm going to put --
slip in the X that you tell me
446
00:29:01,360 --> 00:29:05,500
and I'm going to get zero.
447
00:29:05,500 --> 00:29:09,000
What X would do that job?
448
00:29:09,000 --> 00:29:11,370
Three and negative one?
449
00:29:11,370 --> 00:29:13,610
Is that the one you
picked, or -- yeah.
450
00:29:13,610 --> 00:29:18,720
Or another -- well, if
you picked zero with zero,
451
00:29:18,720 --> 00:29:21,060
I'm not so excited, right?
452
00:29:21,060 --> 00:29:23,490
Because that would always work.
453
00:29:23,490 --> 00:29:26,550
So it's really the
fact that this vector
454
00:29:26,550 --> 00:29:28,840
isn't zero that's important.
455
00:29:28,840 --> 00:29:33,750
It's a non-zero vector and
three negative one would do it.
456
00:29:33,750 --> 00:29:37,130
That just says three of this
column minus one of that column
457
00:29:37,130 --> 00:29:38,861
is the zero column.
458
00:29:38,861 --> 00:29:39,360
Okay.
459
00:29:42,240 --> 00:29:46,860
So now I know that A
couldn't be invertible.
460
00:29:46,860 --> 00:29:49,890
But what's the reasoning?
461
00:29:49,890 --> 00:29:54,720
If A X is zero, suppose I
multiplied by A inverse.
462
00:29:54,720 --> 00:29:56,660
Yeah, well here's the reason.
463
00:29:56,660 --> 00:30:02,180
Here -- this is why this
spells disaster for an inverse.
464
00:30:02,180 --> 00:30:06,580
The matrix can't have an
inverse if some combination
465
00:30:06,580 --> 00:30:09,140
of the columns gives
z- it gives nothing.
466
00:30:09,140 --> 00:30:12,630
Because, I could
take A X equals zero,
467
00:30:12,630 --> 00:30:21,200
I could multiply by A inverse
and what would I discover?
468
00:30:21,200 --> 00:30:23,840
Suppose I take that equation
and I multiply by --
469
00:30:23,840 --> 00:30:26,860
if A inverse existed, which
of course I'm going to come
470
00:30:26,860 --> 00:30:30,560
to the conclusion it can't
because if it existed,
471
00:30:30,560 --> 00:30:33,570
if there was an A inverse
to this dopey matrix,
472
00:30:33,570 --> 00:30:36,990
I would multiply that equation
by that inverse and I would
473
00:30:36,990 --> 00:30:42,270
discover X is zero.
474
00:30:42,270 --> 00:30:45,320
If I multiply A by A inverse
on the left, I get X.
475
00:30:45,320 --> 00:30:48,830
If I multiply by A inverse
on the right, I get zero.
476
00:30:48,830 --> 00:30:50,790
So I would discover X was zero.
477
00:30:50,790 --> 00:30:53,050
But it -- X is not zero.
478
00:30:53,050 --> 00:30:54,800
X -- this guy wasn't zero.
479
00:30:54,800 --> 00:30:55,460
There it is.
480
00:30:55,460 --> 00:30:58,170
It's three minus one.
481
00:30:58,170 --> 00:31:06,250
So, conclusion -- only, it takes
us some time to really work
482
00:31:06,250 --> 00:31:08,040
with that conclusion --
483
00:31:08,040 --> 00:31:14,300
our conclusion will be that
non-invertible matrices,
484
00:31:14,300 --> 00:31:19,490
singular matrices, some
combinations of their columns
485
00:31:19,490 --> 00:31:22,290
gives the zero column.
486
00:31:22,290 --> 00:31:26,090
They they take some
vector X into zero.
487
00:31:26,090 --> 00:31:30,650
And there's no way A
inverse can recover, right?
488
00:31:30,650 --> 00:31:32,520
That's what this equation says.
489
00:31:32,520 --> 00:31:36,980
This equation says I take
this vector X and multiplying
490
00:31:36,980 --> 00:31:39,470
by A gives zero.
491
00:31:39,470 --> 00:31:42,220
But then when I
multiply by A inverse,
492
00:31:42,220 --> 00:31:44,690
I can never escape from zero.
493
00:31:44,690 --> 00:31:47,980
So there couldn't
be an A inverse.
494
00:31:47,980 --> 00:31:51,701
Where here -- okay, now fix --
495
00:31:51,701 --> 00:31:52,200
all right.
496
00:31:52,200 --> 00:31:57,200
Now let me take -- all right,
back to the positive side.
497
00:31:57,200 --> 00:32:01,530
Let's take a matrix that
does have an inverse.
498
00:32:01,530 --> 00:32:03,840
And why not invert it?
499
00:32:03,840 --> 00:32:04,500
Okay.
500
00:32:04,500 --> 00:32:08,782
Can I -- so let me take on
this third board a matrix --
501
00:32:08,782 --> 00:32:09,990
shall I fix that up a little?
502
00:32:12,810 --> 00:32:15,660
Tell me a matrix that
has got an inverse.
503
00:32:18,680 --> 00:32:22,930
Well, let me say one three
two -- what shall I put there?
504
00:32:25,620 --> 00:32:28,665
Well, don't put six,
I guess is -- right?
505
00:32:31,570 --> 00:32:35,000
Do I any favorites here?
506
00:32:35,000 --> 00:32:36,910
One?
507
00:32:36,910 --> 00:32:38,280
Or eight?
508
00:32:41,470 --> 00:32:42,140
I don't care.
509
00:32:42,140 --> 00:32:43,450
What, seven?
510
00:32:43,450 --> 00:32:43,950
Seven.
511
00:32:43,950 --> 00:32:44,540
Okay.
512
00:32:44,540 --> 00:32:46,060
Seven is a lucky number.
513
00:32:46,060 --> 00:32:48,750
All right, seven, okay.
514
00:32:48,750 --> 00:32:49,380
Okay.
515
00:32:49,380 --> 00:32:51,080
So -- now what's our idea?
516
00:32:51,080 --> 00:32:53,740
We believe that this
matrix is invertible.
517
00:32:53,740 --> 00:32:57,060
Those who like determinants have
quickly taken its determinant
518
00:32:57,060 --> 00:32:59,320
and found it wasn't zero.
519
00:32:59,320 --> 00:33:04,610
Those who like columns,
and probably that --
520
00:33:04,610 --> 00:33:08,450
that department is not
totally popular yet --
521
00:33:08,450 --> 00:33:11,080
but those who like columns
will look at those two columns
522
00:33:11,080 --> 00:33:15,090
and say, hey, they point
in different directions.
523
00:33:15,090 --> 00:33:18,080
So I can get anything.
524
00:33:18,080 --> 00:33:19,860
Now, let me see, what do I mean?
525
00:33:19,860 --> 00:33:21,880
How I going to
computer A inverse?
526
00:33:21,880 --> 00:33:24,060
So A inverse --
527
00:33:24,060 --> 00:33:28,360
here's A inverse, now,
and I have to find it.
528
00:33:28,360 --> 00:33:33,219
And what do I get when I
do this multiplication?
529
00:33:33,219 --> 00:33:33,760
The identity.
530
00:33:40,810 --> 00:33:43,940
You know, forgive me for
taking two by two-s, but --
531
00:33:43,940 --> 00:33:49,240
lt's good to keep the
computations manageable and let
532
00:33:49,240 --> 00:33:51,090
the ideas come out.
533
00:33:51,090 --> 00:33:55,230
Okay, now what's
the idea I want?
534
00:33:55,230 --> 00:33:57,520
I'm looking for this
matrix A inverse, how
535
00:33:57,520 --> 00:33:58,750
I going to find it?
536
00:33:58,750 --> 00:34:04,235
Right now, I've got
four numbers to find.
537
00:34:08,290 --> 00:34:12,159
I'm going to look
at the first column.
538
00:34:12,159 --> 00:34:16,860
Let me take this
first column, A B.
539
00:34:16,860 --> 00:34:18,699
What's up there?
540
00:34:18,699 --> 00:34:20,670
What -- tell me this.
541
00:34:20,670 --> 00:34:25,260
What equation does the
first column satisfy?
542
00:34:25,260 --> 00:34:31,489
The first column satisfies A
times that column is one zero.
543
00:34:31,489 --> 00:34:34,179
The first column of the answer.
544
00:34:34,179 --> 00:34:38,820
And the second column,
C D, satisfies A times
545
00:34:38,820 --> 00:34:41,139
that second column is zero one.
546
00:34:41,139 --> 00:34:48,909
You see that finding the inverse
is like solving two systems.
547
00:34:48,909 --> 00:34:52,210
One system, when the
right-hand side is one zero --
548
00:34:52,210 --> 00:34:54,139
I'm just going to split
it into two pieces.
549
00:34:57,427 --> 00:34:58,760
I don't even need to rewrite it.
550
00:34:58,760 --> 00:35:04,480
I can take A times --
so let me put it here.
551
00:35:04,480 --> 00:35:18,100
A times column j of A inverse
is column j of the identity.
552
00:35:21,350 --> 00:35:23,010
I've got n equations.
553
00:35:23,010 --> 00:35:26,500
I've got, well,
two in this case.
554
00:35:26,500 --> 00:35:29,100
And they have the
same matrix, A,
555
00:35:29,100 --> 00:35:30,860
but they have different
right-hand sides.
556
00:35:30,860 --> 00:35:32,840
The right-hand sides
are just the columns
557
00:35:32,840 --> 00:35:35,800
of the identity, this
guy and this guy.
558
00:35:35,800 --> 00:35:37,450
And these are the two solutions.
559
00:35:37,450 --> 00:35:39,610
Do you see what I'm going --
560
00:35:39,610 --> 00:35:45,120
I'm looking at that
equation by columns.
561
00:35:45,120 --> 00:35:47,050
I'm looking at A
times this column,
562
00:35:47,050 --> 00:35:49,910
giving that guy, and A times
that column giving that guy.
563
00:35:49,910 --> 00:35:55,500
So -- Essentially -- so
this is like the Gauss --
564
00:35:55,500 --> 00:35:56,410
we're back to Gauss.
565
00:35:56,410 --> 00:36:00,820
We're back to solving systems of
equations, but we're solving --
566
00:36:00,820 --> 00:36:05,160
we've got two right-hand
sides instead of one.
567
00:36:05,160 --> 00:36:07,010
That's where Jordan comes in.
568
00:36:07,010 --> 00:36:09,980
So at the very beginning
of the lecture,
569
00:36:09,980 --> 00:36:13,230
I mentioned Gauss-Jordan,
let me write it up again.
570
00:36:13,230 --> 00:36:13,730
Okay.
571
00:36:13,730 --> 00:36:16,630
Here's the Gauss-Jordan idea.
572
00:36:21,600 --> 00:36:35,000
Gauss-Jordan solve
two equations at once.
573
00:36:39,460 --> 00:36:39,960
Okay.
574
00:36:39,960 --> 00:36:43,630
Let me show you how
the mechanics go.
575
00:36:43,630 --> 00:36:48,970
How do I solve a
single equation?
576
00:36:48,970 --> 00:36:55,760
So the two equations
are one three two seven,
577
00:36:55,760 --> 00:37:01,630
multiplying A B gives one zero.
578
00:37:01,630 --> 00:37:03,560
And the other
equation is the same
579
00:37:03,560 --> 00:37:12,130
one three two seven
multiplying C D gives zero one.
580
00:37:12,130 --> 00:37:15,010
Okay.
581
00:37:15,010 --> 00:37:17,410
That'll tell me the two
columns of the inverse.
582
00:37:17,410 --> 00:37:19,080
I'll have inverse.
583
00:37:19,080 --> 00:37:22,940
In other words, if I can
solve with this matrix A,
584
00:37:22,940 --> 00:37:24,850
if I can solve with
that right-hand side
585
00:37:24,850 --> 00:37:28,340
and that right-hand
side, I'm invertible.
586
00:37:28,340 --> 00:37:29,460
I've got it.
587
00:37:29,460 --> 00:37:30,690
Okay.
588
00:37:30,690 --> 00:37:36,410
And Jordan sort of said to
Gauss, solve them together,
589
00:37:36,410 --> 00:37:39,650
look at the matrix -- if
we just solve this one,
590
00:37:39,650 --> 00:37:43,140
I would look at one
three two seven,
591
00:37:43,140 --> 00:37:46,030
and how do I deal with
the right-hand side?
592
00:37:46,030 --> 00:37:48,960
I stick it on as an
extra column, right?
593
00:37:56,130 --> 00:37:58,460
That's this augmented matrix.
594
00:37:58,460 --> 00:38:01,750
That's the matrix when I'm
watching the right-hand side
595
00:38:01,750 --> 00:38:04,470
at the same time, doing the
same thing to the right side
596
00:38:04,470 --> 00:38:06,520
that I do to the left?
597
00:38:06,520 --> 00:38:09,610
So I just carry it along
as an extra column.
598
00:38:09,610 --> 00:38:12,100
Now I'm going to carry
along two extra columns.
599
00:38:17,200 --> 00:38:20,920
And I'm going to do
whatever Gauss wants, right?
600
00:38:20,920 --> 00:38:24,150
I'm going to do elimination.
601
00:38:24,150 --> 00:38:27,010
I'm going to get
this to be simple
602
00:38:27,010 --> 00:38:30,070
and this thing will
turn into the inverse.
603
00:38:30,070 --> 00:38:32,460
This is what's coming.
604
00:38:32,460 --> 00:38:35,800
I'm going to do elimination
steps to make this
605
00:38:35,800 --> 00:38:39,240
into the identity,
and lo and behold,
606
00:38:39,240 --> 00:38:41,550
the inverse will show up here.
607
00:38:41,550 --> 00:38:43,330
K--- let's do it.
608
00:38:43,330 --> 00:38:43,830
Okay.
609
00:38:43,830 --> 00:38:46,880
So what are the
elimination steps?
610
00:38:46,880 --> 00:38:51,680
So you see -- here's my matrix
A and here's the identity, like,
611
00:38:51,680 --> 00:38:52,795
stuck on, augmented on.
612
00:38:52,795 --> 00:38:53,670
STUDENT: I'm sorry...
613
00:38:53,670 --> 00:38:54,211
STRANG: Yeah?
614
00:38:54,211 --> 00:38:58,800
STUDENT: -- is the two and the
three supposed to be switched?
615
00:38:58,800 --> 00:39:01,070
STRANG: Did I -- oh, no,
they weren't supposed to be
616
00:39:01,070 --> 00:39:01,570
switched.
617
00:39:01,570 --> 00:39:02,070
Sorry.
618
00:39:02,070 --> 00:39:04,680
Thanks.
619
00:39:04,680 --> 00:39:05,930
Okay.
620
00:39:05,930 --> 00:39:08,425
Thank you very much.
621
00:39:08,425 --> 00:39:09,800
And there -- I've
got them right.
622
00:39:09,800 --> 00:39:11,215
Okay, thanks.
623
00:39:14,520 --> 00:39:15,510
Okay.
624
00:39:15,510 --> 00:39:17,780
So let's do elimination.
625
00:39:17,780 --> 00:39:19,890
All right, it's going
to be simple, right?
626
00:39:19,890 --> 00:39:23,690
So I take two of this
row away from this row.
627
00:39:23,690 --> 00:39:27,180
So this row stays
the same and two
628
00:39:27,180 --> 00:39:28,680
of those come away from this.
629
00:39:28,680 --> 00:39:32,560
That leaves me with a zero and
a one and two of these away from
630
00:39:32,560 --> 00:39:36,970
this is that what
you're getting --
631
00:39:36,970 --> 00:39:39,930
after one elimination step --
632
00:39:39,930 --> 00:39:42,790
Let me sort of separate the --
633
00:39:42,790 --> 00:39:45,040
the left half from
the right half.
634
00:39:45,040 --> 00:39:48,690
So two of that first row got
subtracted from the second row.
635
00:39:48,690 --> 00:39:53,240
Now this is an upper
triangular form.
636
00:39:53,240 --> 00:39:57,120
Gauss would quit, but
Jordan says keeps going.
637
00:39:57,120 --> 00:39:59,250
Use elimination upwards.
638
00:39:59,250 --> 00:40:03,710
Subtract a multiple of
equation two from equation one
639
00:40:03,710 --> 00:40:05,970
to get rid of the three.
640
00:40:05,970 --> 00:40:08,970
So let's go the whole way.
641
00:40:08,970 --> 00:40:14,420
So now I'm going to -- this guy
is fine, but I'm going to --
642
00:40:14,420 --> 00:40:15,770
what do I do now?
643
00:40:15,770 --> 00:40:19,850
What's my final step that
produces the inverse?
644
00:40:19,850 --> 00:40:22,200
I multiply this by
the right number
645
00:40:22,200 --> 00:40:26,060
to get up to ther to
remove that three.
646
00:40:26,060 --> 00:40:28,350
So I guess, I --
since this is a one,
647
00:40:28,350 --> 00:40:30,830
there's the pivot sitting there.
648
00:40:30,830 --> 00:40:33,750
I multiply it by three
and subtract from that,
649
00:40:33,750 --> 00:40:35,080
so what do I get?
650
00:40:35,080 --> 00:40:38,460
I'll have one zero -- oh,
yeah that was my whole point.
651
00:40:38,460 --> 00:40:41,640
I'll multiply this by three
and subtract from that,
652
00:40:41,640 --> 00:40:46,450
which will give me seven.
653
00:40:46,450 --> 00:40:49,340
And I multiply this by three
and subtract from that,
654
00:40:49,340 --> 00:40:50,920
which gives me a minus three.
655
00:41:00,060 --> 00:41:06,500
And what's my hope, belief?
656
00:41:06,500 --> 00:41:10,780
Here I started with
A and the identity,
657
00:41:10,780 --> 00:41:16,410
and I ended up with
the identity and who?
658
00:41:16,410 --> 00:41:18,640
That better be A inverse.
659
00:41:24,090 --> 00:41:26,980
That's the Gauss Jordan idea.
660
00:41:26,980 --> 00:41:33,690
Start with this long matrix,
double-length A I, eliminate,
661
00:41:33,690 --> 00:41:37,450
eliminate until this
part is down to I,
662
00:41:37,450 --> 00:41:40,530
then this one will --
must be for some reason,
663
00:41:40,530 --> 00:41:45,230
and we've got to find the
reason -- must be A inverse.
664
00:41:45,230 --> 00:41:46,690
Shall I just check
that it works?
665
00:41:50,160 --> 00:41:55,080
Let me just check that -- can I
multiply this matrix this part
666
00:41:55,080 --> 00:42:00,300
times A, I'll carry A
over here and just do that
667
00:42:00,300 --> 00:42:01,190
multiplication.
668
00:42:01,190 --> 00:42:03,310
You'll see I'll do it
the old fashioned way.
669
00:42:03,310 --> 00:42:06,000
Seven minus six is a one.
670
00:42:06,000 --> 00:42:08,670
Twenty one minus
twenty one is a zero,
671
00:42:08,670 --> 00:42:13,280
minus two plus two is a zero,
minus six plus seven is a one.
672
00:42:13,280 --> 00:42:15,780
Check.
673
00:42:15,780 --> 00:42:18,290
So that is the inverse.
674
00:42:18,290 --> 00:42:20,950
That's the Gauss-Jordan idea.
675
00:42:20,950 --> 00:42:24,190
So, you'll -- one of the
homework problems or more than
676
00:42:24,190 --> 00:42:30,710
one for Wednesday will ask
you to go through those steps.
677
00:42:30,710 --> 00:42:33,560
I think you just got to go
through Gauss-Jordan a couple
678
00:42:33,560 --> 00:42:38,600
of times, but I --
679
00:42:38,600 --> 00:42:43,690
yeah -- just to
see the mechanics.
680
00:42:43,690 --> 00:42:48,000
But the, important
thing is, why --
681
00:42:48,000 --> 00:42:50,190
is, like, what happened?
682
00:42:50,190 --> 00:42:53,570
Why did we -- why did
we get A inverse there?
683
00:42:53,570 --> 00:42:54,690
Let me ask you that.
684
00:42:57,860 --> 00:43:03,190
We got -- so we take --
685
00:43:03,190 --> 00:43:08,020
We do row reduction, we do
elimination on this long matrix
686
00:43:08,020 --> 00:43:12,840
A I until the first half
687
00:43:12,840 --> 00:43:14,990
Then a second half
is A inverse. is up.
688
00:43:14,990 --> 00:43:20,050
Well, how do I see that?
689
00:43:20,050 --> 00:43:22,310
Let me put up here
how I see that.
690
00:43:22,310 --> 00:43:30,100
So here's my Gauss-Jordan thing,
and I'm doing stuff to it.
691
00:43:30,100 --> 00:43:34,320
So I'm -- well,
whole lot of E's.
692
00:43:37,530 --> 00:43:40,150
Remember those are those
elimination matrices.
693
00:43:40,150 --> 00:43:42,861
Those are the -- those are the
things that we figured out last
694
00:43:42,861 --> 00:43:43,360
time.
695
00:43:43,360 --> 00:43:49,370
Yes, that's what an elimination
step is it's in matrix form,
696
00:43:49,370 --> 00:43:51,430
I'm multiplying by some Es.
697
00:43:51,430 --> 00:43:55,120
And the result -- well, so I'm
multiplying by a whole bunch
698
00:43:55,120 --> 00:43:55,650
of Es.
699
00:43:55,650 --> 00:43:57,530
So, I get a --
700
00:43:57,530 --> 00:44:02,070
can I call the overall matrix E?
701
00:44:02,070 --> 00:44:06,420
That's the elimination matrix,
the product of all those little
702
00:44:06,420 --> 00:44:06,920
pieces.
703
00:44:06,920 --> 00:44:09,050
What do I mean by little pieces?
704
00:44:09,050 --> 00:44:11,160
Well, there was an
elimination matrix
705
00:44:11,160 --> 00:44:14,400
that subtracted two of
that away from that.
706
00:44:14,400 --> 00:44:16,180
Then there was an
elimination matrix
707
00:44:16,180 --> 00:44:19,160
that subtracted three
of that away from that.
708
00:44:19,160 --> 00:44:21,370
I guess in this
case, that was all.
709
00:44:21,370 --> 00:44:24,280
So there were just two
Es in this case, one
710
00:44:24,280 --> 00:44:26,630
that did this step and
one that did this step
711
00:44:26,630 --> 00:44:31,690
and together they gave me
an E that does both steps.
712
00:44:31,690 --> 00:44:38,560
And the net result
was to get an I here.
713
00:44:38,560 --> 00:44:45,490
And you can tell me
what that has to be.
714
00:44:45,490 --> 00:44:50,240
This is, like, the
picture of what happened.
715
00:44:50,240 --> 00:44:53,350
If E multiplied A,
whatever that E is --
716
00:44:53,350 --> 00:44:59,780
we never figured
it out in this way.
717
00:44:59,780 --> 00:45:06,692
But whatever that E times
that E is, E times A is --
718
00:45:09,410 --> 00:45:12,120
What's E times A?
719
00:45:12,120 --> 00:45:13,480
It's I.
720
00:45:13,480 --> 00:45:20,510
That E, whatever the heck it
was, multiplied A and produced
721
00:45:20,510 --> 00:45:22,710
So E must be --
722
00:45:22,710 --> 00:45:30,640
E A equaling I tells us what
E is, I. namely it is --
723
00:45:30,640 --> 00:45:32,110
STUDENT: It's the inverse of A.
724
00:45:32,110 --> 00:45:33,720
STRANG: It's the inverse of A.
725
00:45:33,720 --> 00:45:35,660
Great.
726
00:45:35,660 --> 00:45:39,780
And therefore, when the second
half, when E multiplies I,
727
00:45:39,780 --> 00:45:41,470
it's E --
728
00:45:41,470 --> 00:45:44,760
Put this A inverse.
729
00:45:44,760 --> 00:45:48,450
You see the picture
looking that way?
730
00:45:48,450 --> 00:45:49,910
E times A is the identity.
731
00:45:49,910 --> 00:45:53,480
It tells us what E has to be.
732
00:45:53,480 --> 00:45:55,600
It has to be the
inverse, and therefore,
733
00:45:55,600 --> 00:45:57,870
on the right-hand
side, where E --
734
00:45:57,870 --> 00:46:00,580
where we just smartly
tucked on the identity,
735
00:46:00,580 --> 00:46:03,060
it's turning in, step by step --
736
00:46:03,060 --> 00:46:05,870
It's turning into A inverse.
737
00:46:05,870 --> 00:46:12,410
There is the statement of
Gauss-Jordan elimination.
738
00:46:12,410 --> 00:46:14,590
That's how you find the inverse.
739
00:46:14,590 --> 00:46:19,210
Where we can look at
it as elimination,
740
00:46:19,210 --> 00:46:24,260
as solving n equations
at the same time -- --
741
00:46:24,260 --> 00:46:28,170
and tacking on n columns,
solving those equations and up
742
00:46:28,170 --> 00:46:32,740
goes the n columns of
A inverse Okay, thanks.
743
00:46:32,740 --> 00:46:34,640
See you on Wednesday.