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Okay.
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This is lecture five
in linear algebra.
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And, it will complete
this chapter of the book.
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So the last section
of this chapter
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is two point seven that talks
about permutations, which
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finished the previous
lecture, and transposes,
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which also came in
the previous lecture.
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There's a little more to do
with those guys, permutations
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and transposes.
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But then the heart of the
lecture will be the beginning
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of what you could say is the
beginning of linear algebra,
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the beginning of real linear
algebra which is seeing
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a bigger picture with vector
spaces -- not just vectors,
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but spaces of vectors and
sub-spaces of those spaces.
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So we're a little ahead
of the syllabus, which
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is good, because we're
coming to the place
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where, there's a lot to
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do.
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Okay.
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So, to begin with permutations.
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Can I just --
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so these permutations, those
are matrices P and they execute
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row exchanges.
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And we may need them.
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We may have a
perfectly good matrix,
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a perfect matrix A that's
invertible that we can solve A
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x=b, but to do it --
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I've got to allow myself
that extra freedom
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that if a zero shows up in the
pivot position I move it away.
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I get a non-zero.
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I get a proper pivot there by
exchanging from a row below.
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And you've seen that
already, and I just
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want to collect
the ideas together.
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And principle, I
could even have to do
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that two times, or more times.
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So I have to allow --
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to complete the --
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the theory, the possibility
that I take my matrix A,
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I start elimination, I find
out that I need row exchanges
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and I do it and
continue and I finish.
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Okay.
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Then all I want to do is say --
and I won't make a big project
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out of this --
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what happens to A equal L U?
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So A equal L U --
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this was a matrix L with ones
on the diagonal and zeroes
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above and multipliers
below, and this U
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we know, with zeroes down here.
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That's only possible.
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That description of
elimination assumes
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that we don't have a P, that we
don't have any row exchanges.
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And now I just want
to say, okay, how
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do I account for row exchanges?
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Because that doesn't.
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The P in this factorization
is the identity matrix.
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The rows were in a good
order, we left them there.
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Maybe I'll just add a
little moment of reality,
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too, about how Matlab
actually does elimination.
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Matlab not only checks whether
that pivot is not zero,
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as every human would do.
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It checks for is that
pivot big enough,
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because it doesn't like
very, very small pivots.
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Pivots close to zero
are numerically bad.
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So actually if we ask
Matlab to solve a system,
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it will do some elimination
some row exchanges, which
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we don't think are necessary.
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Algebra doesn't say they're
necessary, but accuracy --
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numerical accuracy
says they are.
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Well, we're doing
algebra, so here we
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will say, well, what
do row exchanges do,
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but we won't do them
unless we have to.
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But we may have to.
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And then, the result is --
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it's hiding here.
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It's the main fact.
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This is the description of
elimination with row exchanges.
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So A equal L U
becomes P A equal L U.
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So this P is the matrix
that does the row exchanges,
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and actually it does them --
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it gets the rows
into the right order,
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into the good order
where pivots will not --
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where zeroes won't appear
in the pivot position,
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where L and U will come
out right as up here.
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So, that's the point.
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Actually, I don't want
to labor that point,
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that a permutation matrix --
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and you remember
what those were.
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I'll remind you from last time
of what the main points about
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permutation matrices were --
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and then just leave
this factorization
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as the general case.
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This is -- any
invertible A we get this.
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For almost every one,
we don't need a P.
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But there's that handful
that do need row exchanges,
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and if we do need
them, there they are.
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Okay, finally, just to
remember what P was.
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So permutations, P is
the identity matrix
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with reordered rows.
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I include in reordering the
possibility that you just
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leave them the same.
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So the identity
matrix is -- okay.
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That's, like, your basic
permutation matrix --
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your do-nothing permutation
matrix is the identity.
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And then there are the ones
that exchange two rows and then
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the ones that exchange three
rows and then then ones that
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exchange four --
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well, it gets a little --
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it gets more interesting
algebraically
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if you've got four rows,
you might exchange them
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all in one big cycle.
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One to two, two to three,
three to four, four to one.
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Or you might have -- exchange
one and two and three and four.
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Lots of possibilities there.
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In fact, how many possibilities?
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The answer was (n)factorial.
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This is n(n-1)(n-2)...
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(3)(2)(1).
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That's the number of --
this counts the reorderings,
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the possible reorderings.
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So it counts all the
n by n permutations.
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And all those
matrices have these --
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have this nice property
that they're all invertible,
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because we can bring those rows
back into the normal order.
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And the matrix that
does that is just P --
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is just the same
as the transpose.
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You might take a
permutation matrix,
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multiply by its transpose
and you will see how --
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that the ones hit the ones and
give the ones in the identity
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matrix.
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So this is a --
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we'll be highly
interested in matrices
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that have nice properties.
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And one property that -- maybe
I could rewrite that as P
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transpose P is the identity.
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That tells me in
other words that this
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is the inverse of that.
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Okay.
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We'll be interested in
matrices that have P transpose
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P equal the identity.
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There are more of them
than just permutations,
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but my point right now is that
permutations are like a little
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group in the middle --
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in the center of these
special matrices.
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Okay.
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So now we know how
many there are.
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Twenty four in the case of
-- there are twenty four four
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by four permutations, there
are five factorial which is
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a hundred and twenty, five times
twenty four would bump us up
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to a hundred and twenty -- so
listing all the five by five
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permutations would
be not so much fun.
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Okay.
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So that's permutations.
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Now also in section two seven is
some discussion of transposes.
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And can I just complete
that discussion.
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First of all, I haven't
even transposed a matrix
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on the board here, have I?
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So I'd better do it.
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So suppose I take a matrix
like (1 2 4; 3 3 1).
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It's a rectangular
matrix, three by two.
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And I want to transpose it.
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So what's --
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I'll use a T, also
Matlab would use a prime.
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And the result will be --
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I'll right it here, because this
was three rows and two columns,
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this was a three by two matrix.
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The transpose will be two
rows and three columns,
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two by three.
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So it's short and wider.
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And, of course, that row --
that column becomes a row --
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that column becomes
the other row.
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And at the same time,
that row became a column.
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This row became a column.
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Oh, what's the general
formula for the transpose?
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So the transpose --
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you see it in numbers.
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What I'm going to write is
the same thing in symbols.
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The numbers are the
clearest, of course.
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But in symbols, if
I take A transpose
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and I ask what number is in row
I and column J of A transpose?
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Well, it came out of A.
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It came out A by this flip
across the main diagonal.
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And, actually, it
was the number in A
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which was in row J, column I.
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So the row and column --
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the row and column
numbers just get reversed.
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The row number becomes
the column number,
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the column number
becomes the row number.
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No problem.
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Okay.
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Now, a special --
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the best matrices, we could say.
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In a lot of applications,
symmetric matrices show up.
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So can I just call attention
to symmetric matrices?
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What does that mean?
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What does that word
symmetric mean?
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It means that this transposing
doesn't change the matrix.
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A transpose equals A.
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And an example.
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So, let's take a matrix
that's symmetric,
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so whatever is sitting
on the diagonal --
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but now what's above the
diagonal, like a one,
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had better be there, a
seven had better be here,
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a nine had better be there.
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There's a symmetric matrix.
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I happened to use all positive
numbers as its entries.
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That's not the point.
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The point is that if I
transpose that matrix,
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I get it back again.
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So symmetric matrices have this
property A transpose equals A.
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I guess at this point --
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I'm just asking you to notice
this family of matrices that
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are unchanged by transposing.
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00:14:17,830 --> 00:14:22,950
And they're easy to
identify, of course.
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00:14:22,950 --> 00:14:28,060
You know, it's not maybe so
easy before we had a case where
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00:14:28,060 --> 00:14:31,830
the transpose gave the inverse.
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00:14:31,830 --> 00:14:36,230
That's highly important,
but not so simple to see.
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This is the case where the
transpose gives the same matrix
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00:14:39,200 --> 00:14:39,720
back again.
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That's totally simple to see.
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Okay.
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00:14:43,380 --> 00:14:49,410
Could actually -- maybe I could
even say when would we get such
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00:14:49,410 --> 00:14:51,150
a matrix?
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00:14:51,150 --> 00:14:55,740
For example, this -- that
matrix is absolutely far from
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symmetric, right?
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00:14:57,850 --> 00:15:01,150
The transpose isn't
even the same shape --
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because it's rectangular,
it turns the --
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lies down on its side.
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But let me tell you a way to
get a symmetric matrix out of
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this.
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Multiply those together.
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If I multiply this
rectangular, shall I
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call it R for rectangular?
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So let that be R for
rectangular matrix
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and let that be R
transpose, which it is.
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Then I think that if I
multiply those together,
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I get a symmetric matrix.
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Can I just do it
with the numbers
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and then ask you why, how did
I know it would be symmetric?
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So my point is that R transpose
R is always symmetric.
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Okay?
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00:15:57,560 --> 00:16:01,470
And I'm going to do it for that
particular R transpose R which
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00:16:01,470 --> 00:16:02,750
was --
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00:16:02,750 --> 00:16:05,815
let's see, the column was
one two four three three one.
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I called that one R
transpose, didn't I,
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and I called this guy one
two four three three one.
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I called that R.
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Shall we just do
that multiplication?
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00:16:19,970 --> 00:16:22,960
Okay, so up here
I'm getting a ten.
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Next to it I'm getting two, a
nine, I'm getting an eleven.
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00:16:27,310 --> 00:16:30,390
Next to that I'm getting
four and three, a seven.
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00:16:30,390 --> 00:16:32,650
Now what do I get there?
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00:16:32,650 --> 00:16:37,650
This eleven came from one
three times two three, right?
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00:16:37,650 --> 00:16:39,720
Row one, column two.
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00:16:39,720 --> 00:16:41,530
What goes here?
255
00:16:41,530 --> 00:16:43,360
Row two, column one.
256
00:16:43,360 --> 00:16:46,050
But no difference.
257
00:16:46,050 --> 00:16:50,220
One three two three or two
three one three, same thing.
258
00:16:50,220 --> 00:16:51,750
It's going to be an eleven.
259
00:16:51,750 --> 00:16:54,310
That's the symmetry.
260
00:16:54,310 --> 00:16:56,230
I can continue to fill it out.
261
00:16:56,230 --> 00:16:58,100
What -- oh, let's
get that seven.
262
00:16:58,100 --> 00:17:00,520
That seven will show
up down here, too,
263
00:17:00,520 --> 00:17:02,650
and then four more numbers.
264
00:17:02,650 --> 00:17:06,619
That seven will show up here
because one three times four
265
00:17:06,619 --> 00:17:10,630
one gave the seven, but also
four one times one three
266
00:17:10,630 --> 00:17:11,599
will give that seven.
267
00:17:11,599 --> 00:17:16,030
Do you see that it works?
268
00:17:16,030 --> 00:17:24,630
Actually, do you want to see it
work also in matrix language?
269
00:17:24,630 --> 00:17:27,089
I mean, that's quite
convincing, right?
270
00:17:27,089 --> 00:17:29,965
That seven is no accident.
271
00:17:32,630 --> 00:17:35,550
The eleven is no accident.
272
00:17:35,550 --> 00:17:40,680
But just tell me how do I know
if I transpose this guy --
273
00:17:40,680 --> 00:17:42,820
How do I know it's symmetric?
274
00:17:42,820 --> 00:17:45,620
Well, I'm going to transpose it.
275
00:17:45,620 --> 00:17:49,610
And when I transpose
it, I'm hoping
276
00:17:49,610 --> 00:17:52,210
I get the matrix back again.
277
00:17:52,210 --> 00:17:54,630
So can I transpose
R transpose R?
278
00:17:54,630 --> 00:17:56,157
So just -- so, why?
279
00:17:59,880 --> 00:18:02,945
Well, my suggestion
is take the transpose.
280
00:18:09,800 --> 00:18:11,670
That's the only way to
show it's symmetric.
281
00:18:11,670 --> 00:18:14,340
Take the transpose and
see that it didn't change.
282
00:18:14,340 --> 00:18:19,970
Okay, so I take the
transpose of R transpose R.
283
00:18:19,970 --> 00:18:20,470
Okay.
284
00:18:20,470 --> 00:18:23,190
How do I do that?
285
00:18:23,190 --> 00:18:27,680
This is our little practice
on the rules for transposes.
286
00:18:27,680 --> 00:18:33,780
So the rule for transposes
is the order gets reversed.
287
00:18:33,780 --> 00:18:38,300
Just like inverses,
which we did prove,
288
00:18:38,300 --> 00:18:44,390
same rule for transposes
and -- which we'll now use.
289
00:18:44,390 --> 00:18:46,110
So the order gets reversed.
290
00:18:46,110 --> 00:18:51,600
It's the transpose of
that that comes first,
291
00:18:51,600 --> 00:18:56,880
and the transpose of
this that comes -- no.
292
00:18:56,880 --> 00:18:58,010
Is that -- yeah.
293
00:18:58,010 --> 00:19:01,130
That's what I have
to write, right?
294
00:19:01,130 --> 00:19:03,500
This is a product of two
matrices and I want its
295
00:19:03,500 --> 00:19:04,350
transpose.
296
00:19:04,350 --> 00:19:06,880
So I put the matrices
in the opposite order
297
00:19:06,880 --> 00:19:08,480
and I transpose them.
298
00:19:08,480 --> 00:19:09,780
But what have I got here?
299
00:19:09,780 --> 00:19:12,320
What is R transpose transpose?
300
00:19:15,300 --> 00:19:18,670
Well, don't all speak at once.
301
00:19:18,670 --> 00:19:22,080
R transpose transpose, I
flipped over the diagonal,
302
00:19:22,080 --> 00:19:28,540
I flipped over the diagonal
again, so I've got R.
303
00:19:28,540 --> 00:19:32,770
And that's just my point, that
if I started with this matrix,
304
00:19:32,770 --> 00:19:34,740
I transposed it, I
got it back again.
305
00:19:37,850 --> 00:19:46,810
So that's the check, without
using numbers, but with --
306
00:19:46,810 --> 00:19:52,510
it checked in two lines that I
always get symmetric matrices
307
00:19:52,510 --> 00:19:53,480
this way.
308
00:19:53,480 --> 00:19:55,680
And actually, that's
where they come
309
00:19:55,680 --> 00:20:00,220
from in so many
practical applications.
310
00:20:00,220 --> 00:20:02,730
Okay.
311
00:20:02,730 --> 00:20:07,580
So now I've said something today
about permutations and about
312
00:20:07,580 --> 00:20:14,540
transposes and about
symmetry and I'm ready
313
00:20:14,540 --> 00:20:17,290
for chapter three.
314
00:20:17,290 --> 00:20:21,550
Can we take a breath --
315
00:20:21,550 --> 00:20:25,250
the tape won't take a breath,
but the lecturer will,
316
00:20:25,250 --> 00:20:31,980
because to tell you
about vector spaces is --
317
00:20:31,980 --> 00:20:38,220
we really have to start now
and think, okay, listen up.
318
00:20:38,220 --> 00:20:39,500
What are vector spaces?
319
00:20:47,400 --> 00:20:48,400
And what are sub-spaces?
320
00:20:51,160 --> 00:20:51,890
Okay.
321
00:20:51,890 --> 00:21:01,430
So, the point is, The main
operations that we do --
322
00:21:01,430 --> 00:21:04,450
what do we do with vectors?
323
00:21:04,450 --> 00:21:05,820
We add them.
324
00:21:05,820 --> 00:21:07,270
We know how to add two vectors.
325
00:21:09,880 --> 00:21:13,930
We multiply them by numbers,
usually called scalers.
326
00:21:13,930 --> 00:21:17,350
If we have a vector, we
know what three V is.
327
00:21:17,350 --> 00:21:24,030
If we have a vector V and
W, we know what V plus W is.
328
00:21:24,030 --> 00:21:26,670
Those are the two
operations that we've
329
00:21:26,670 --> 00:21:29,090
got to be able to do.
330
00:21:29,090 --> 00:21:33,220
To legitimately talk
about a space of vectors,
331
00:21:33,220 --> 00:21:35,870
the requirement
is that we should
332
00:21:35,870 --> 00:21:39,990
be able to add the things
and multiply by numbers
333
00:21:39,990 --> 00:21:44,380
and that there should be
some decent rules satisfied.
334
00:21:44,380 --> 00:21:45,370
Okay.
335
00:21:45,370 --> 00:21:48,590
So let me start with examples.
336
00:21:48,590 --> 00:21:50,755
So I'm talking now
about vector spaces.
337
00:21:56,810 --> 00:21:58,685
And I'm going to
start with examples.
338
00:22:06,690 --> 00:22:09,480
Let me say again what this
word space is meaning.
339
00:22:09,480 --> 00:22:14,010
When I say that word
space, that means to me
340
00:22:14,010 --> 00:22:19,440
that I've got a bunch of
vectors, a space of vectors.
341
00:22:19,440 --> 00:22:21,010
But not just any
bunch of vectors.
342
00:22:24,010 --> 00:22:28,720
It has to be a
space of vectors --
343
00:22:28,720 --> 00:22:31,300
has to allow me to do the
operations that vectors
344
00:22:31,300 --> 00:22:32,740
are for.
345
00:22:32,740 --> 00:22:37,110
I have to be able to add
vectors and multiply by numbers.
346
00:22:37,110 --> 00:22:39,560
I have to be able to
take linear combinations.
347
00:22:39,560 --> 00:22:43,120
Well, where did we meet
linear combinations?
348
00:22:43,120 --> 00:22:48,960
We met them back in, say in R^2.
349
00:22:48,960 --> 00:22:51,370
So there's a vector space.
350
00:22:51,370 --> 00:22:54,010
What's that vector space?
351
00:22:54,010 --> 00:22:59,270
So R two is telling me I'm
talking about real numbers
352
00:22:59,270 --> 00:23:01,370
and I'm talking about
two real numbers.
353
00:23:01,370 --> 00:23:11,470
So this is all two
dimensional vectors --
354
00:23:11,470 --> 00:23:16,580
real, such as --
355
00:23:16,580 --> 00:23:18,750
well, I'm not going to
be able to list them all.
356
00:23:18,750 --> 00:23:20,210
But let me put a few down.
357
00:23:20,210 --> 00:23:30,420
|3; 2|, |0;0|, |pi; e|.
358
00:23:30,420 --> 00:23:30,920
So on.
359
00:23:35,450 --> 00:23:39,890
And it's natural -- okay.
360
00:23:39,890 --> 00:23:44,270
Let's see, I guess I
should do algebra first.
361
00:23:44,270 --> 00:23:46,980
Algebra means what can
I do to these vectors?
362
00:23:46,980 --> 00:23:48,130
I can add them.
363
00:23:48,130 --> 00:23:50,520
I can add that to that.
364
00:23:50,520 --> 00:23:51,660
And how do I do it?
365
00:23:51,660 --> 00:23:54,460
A component at a
time, of course.
366
00:23:54,460 --> 00:23:58,240
Three two added to zero
zero gives me, three two.
367
00:23:58,240 --> 00:24:00,190
Sorry about that.
368
00:24:00,190 --> 00:24:05,780
Three two added to pi e gives
me three plus pi, two plus e.
369
00:24:05,780 --> 00:24:07,260
Oh, you know what it does.
370
00:24:07,260 --> 00:24:11,240
And you know the picture
that goes with it.
371
00:24:11,240 --> 00:24:14,830
There's the vector three two.
372
00:24:14,830 --> 00:24:19,520
And often, the
picture has an arrow.
373
00:24:19,520 --> 00:24:22,550
The vector zero zero, which is
a highly important vector --
374
00:24:22,550 --> 00:24:24,610
it's got, like, the
most important here
375
00:24:24,610 --> 00:24:25,930
-- is there.
376
00:24:25,930 --> 00:24:29,840
And of course there's
not much of an arrow.
377
00:24:29,840 --> 00:24:35,110
Pi -- I'll have to remember --
pi is about three and a little
378
00:24:35,110 --> 00:24:37,570
more, e is about two
and a little more.
379
00:24:37,570 --> 00:24:41,090
So maybe there's pi e.
380
00:24:41,090 --> 00:24:44,690
I never drew pi e before.
381
00:24:44,690 --> 00:24:47,030
It's just natural to --
382
00:24:47,030 --> 00:24:55,560
this is the first
component on the horizontal
383
00:24:55,560 --> 00:24:59,470
and this is the
second component,
384
00:24:59,470 --> 00:25:02,010
going up the vertical.
385
00:25:02,010 --> 00:25:02,910
Okay.
386
00:25:02,910 --> 00:25:07,570
And the whole plane is R two.
387
00:25:07,570 --> 00:25:14,980
So R two is, we
could say, the plane.
388
00:25:14,980 --> 00:25:17,710
The xy plane.
389
00:25:17,710 --> 00:25:18,920
That's what everybody thinks.
390
00:25:24,770 --> 00:25:32,800
But the point is it's a vector
space because all those vectors
391
00:25:32,800 --> 00:25:34,140
are in there.
392
00:25:34,140 --> 00:25:37,380
If I removed one of them --
393
00:25:37,380 --> 00:25:39,420
Suppose I removed zero zero.
394
00:25:39,420 --> 00:25:43,480
Suppose I tried to take the --
considered the X Y plane with
395
00:25:43,480 --> 00:25:46,360
a puncture, with
a point removed.
396
00:25:46,360 --> 00:25:47,200
Like the origin.
397
00:25:47,200 --> 00:25:50,470
That would be, like, awful
to take the origin away.
398
00:25:50,470 --> 00:25:52,570
Why is that?
399
00:25:52,570 --> 00:25:54,550
Why do I need the origin there?
400
00:25:54,550 --> 00:25:59,570
Because I have to be allowed --
if I had these other vectors,
401
00:25:59,570 --> 00:26:03,330
I have to be allowed to
multiply three two --
402
00:26:03,330 --> 00:26:05,610
this was three two --
403
00:26:05,610 --> 00:26:09,820
by anything, by any
scaler, including zero.
404
00:26:09,820 --> 00:26:12,020
I've got to be allowed
to multiply by zero
405
00:26:12,020 --> 00:26:15,010
and the result's
got to be there.
406
00:26:15,010 --> 00:26:18,110
I can't do without that point.
407
00:26:18,110 --> 00:26:23,670
And I have to be able to add
three two to the opposite guy,
408
00:26:23,670 --> 00:26:26,800
minus three minus two.
409
00:26:26,800 --> 00:26:29,230
And if I add those I'm
back to the origin again.
410
00:26:29,230 --> 00:26:31,360
No way I can do
without the origin.
411
00:26:31,360 --> 00:26:36,280
Every vector space has got
that zero vector in it.
412
00:26:36,280 --> 00:26:38,650
Okay, that's an
easy vector space,
413
00:26:38,650 --> 00:26:42,520
because we have a
natural picture of it.
414
00:26:42,520 --> 00:26:43,840
Okay.
415
00:26:43,840 --> 00:26:46,260
Similarly easy is R^3.
416
00:26:50,340 --> 00:26:54,630
This would be all -- let
me go up a little here.
417
00:26:54,630 --> 00:26:57,820
This would be --
418
00:26:57,820 --> 00:27:02,670
R three would be all three
dimensional vectors --
419
00:27:02,670 --> 00:27:09,645
or shall I say vectors
with three real components.
420
00:27:14,320 --> 00:27:15,000
Okay.
421
00:27:15,000 --> 00:27:21,030
Let me just to be
sure we're together,
422
00:27:21,030 --> 00:27:23,661
let me take the
vector three two zero.
423
00:27:29,410 --> 00:27:33,390
Is that a vector in R^2 or R^3?
424
00:27:33,390 --> 00:27:38,490
Definitely it's in R^3.
425
00:27:38,490 --> 00:27:40,150
It's got three components.
426
00:27:40,150 --> 00:27:43,040
One of them happens to be zero,
but that's a perfectly okay
427
00:27:43,040 --> 00:27:43,850
number.
428
00:27:43,850 --> 00:27:48,290
So that's a vector in R^3.
429
00:27:48,290 --> 00:27:51,590
We don't want to mix up the --
430
00:27:51,590 --> 00:27:55,090
I mean, keep these vectors
straight and keep R^n straight.
431
00:27:55,090 --> 00:27:57,630
So what's R^n?
432
00:27:57,630 --> 00:27:59,150
R^n.
433
00:27:59,150 --> 00:28:05,990
So this is our big example, is
all vectors with n components.
434
00:28:05,990 --> 00:28:11,170
And I'm making these darn
things column vectors.
435
00:28:11,170 --> 00:28:14,050
Can I try to follow
that convention,
436
00:28:14,050 --> 00:28:17,530
that they'll be column vectors,
and their components should
437
00:28:17,530 --> 00:28:20,690
be real numbers.
438
00:28:20,690 --> 00:28:24,830
Later we'll need complex
numbers and complex vectors,
439
00:28:24,830 --> 00:28:26,721
but much later.
440
00:28:26,721 --> 00:28:27,220
Okay.
441
00:28:27,220 --> 00:28:28,780
So that's a vector space.
442
00:28:31,420 --> 00:28:33,670
Now, let's see.
443
00:28:33,670 --> 00:28:35,910
What do I have to tell
you about vector spaces?
444
00:28:35,910 --> 00:28:44,090
I said the most important thing,
which is that we can add any
445
00:28:44,090 --> 00:28:46,760
two of these and
we -- still in R^2.
446
00:28:46,760 --> 00:28:50,220
We can multiply by any number
and we're still in R^2.
447
00:28:50,220 --> 00:28:53,380
We can take any combination
and we're still in R^2.
448
00:28:53,380 --> 00:28:55,290
And same goes for R^n.
449
00:28:55,290 --> 00:29:02,240
It's -- honesty requires me to
mention that these operations
450
00:29:02,240 --> 00:29:08,300
of adding and multiplying
have to obey a few rules.
451
00:29:08,300 --> 00:29:12,790
Like, we can't just arbitrarily
say, okay, the sum of three two
452
00:29:12,790 --> 00:29:15,610
and pi e is zero zero.
453
00:29:15,610 --> 00:29:18,410
It's not.
454
00:29:18,410 --> 00:29:22,650
The sum of three two and
minus three two is zero zero.
455
00:29:22,650 --> 00:29:27,030
So -- oh, I'm not going
to -- the book, actually,
456
00:29:27,030 --> 00:29:32,420
lists the eight rules that the
addition and multiplication
457
00:29:32,420 --> 00:29:34,680
have to satisfy, but they do.
458
00:29:34,680 --> 00:29:38,810
They certainly satisfy it in
R^n and usually it's not those
459
00:29:38,810 --> 00:29:42,170
eight rules that are in doubt.
460
00:29:42,170 --> 00:29:50,070
What's -- the question is, can
we do those additions and do we
461
00:29:50,070 --> 00:29:51,250
stay in the space?
462
00:29:51,250 --> 00:29:55,580
Let me show you a
case where you can't.
463
00:29:55,580 --> 00:29:59,810
So suppose this is going
to be not a vector space.
464
00:30:05,490 --> 00:30:08,780
Suppose I take the xy
plane -- so there's R^2.
465
00:30:08,780 --> 00:30:11,240
That is a vector space.
466
00:30:11,240 --> 00:30:15,940
Now suppose I just
take part of it.
467
00:30:15,940 --> 00:30:17,670
Just this.
468
00:30:17,670 --> 00:30:22,270
Just this one -- this is one
quarter of the vector space.
469
00:30:24,910 --> 00:30:29,965
All the vectors with positive
or at least not negative
470
00:30:29,965 --> 00:30:30,465
components.
471
00:30:33,070 --> 00:30:37,540
Can I add those safely?
472
00:30:37,540 --> 00:30:38,410
Yes.
473
00:30:38,410 --> 00:30:41,690
If I add a vector
with, like, two --
474
00:30:41,690 --> 00:30:45,030
three two to another
vector like five six,
475
00:30:45,030 --> 00:30:48,950
I'm still up in this quarter,
no problem with adding.
476
00:30:48,950 --> 00:30:54,860
But there's a heck of a problem
with multiplying by scalers,
477
00:30:54,860 --> 00:30:58,690
because there's a lot of
scalers that will take me out
478
00:30:58,690 --> 00:31:02,280
of this quarter plane,
like negative ones.
479
00:31:02,280 --> 00:31:05,820
If I took three two and I
multiplied by minus five,
480
00:31:05,820 --> 00:31:08,240
I'm way down here.
481
00:31:08,240 --> 00:31:12,220
So that's not a vector
space, because it's not --
482
00:31:12,220 --> 00:31:14,250
closed is the right word.
483
00:31:14,250 --> 00:31:17,870
It's not closed
under multiplication
484
00:31:17,870 --> 00:31:19,850
by all real numbers.
485
00:31:22,500 --> 00:31:27,150
So a vector space has to be
closed under multiplication
486
00:31:27,150 --> 00:31:29,010
and addition of vectors.
487
00:31:29,010 --> 00:31:31,680
In other words,
linear combinations.
488
00:31:31,680 --> 00:31:37,560
It -- so, it means that if
I give you a few vectors --
489
00:31:37,560 --> 00:31:39,980
yeah look, here's an
important -- here --
490
00:31:39,980 --> 00:31:42,420
now we're getting to some
really important vector spaces.
491
00:31:42,420 --> 00:31:47,460
Well, R^n -- like, they
are the most important.
492
00:31:47,460 --> 00:31:52,520
But we will be interested in
so- in vector spaces that are
493
00:31:52,520 --> 00:31:55,700
inside R^n.
494
00:31:55,700 --> 00:32:01,790
Vector spaces that follow
the rules, but they --
495
00:32:01,790 --> 00:32:10,140
we don't need all of -- see,
there we started with R^2 here,
496
00:32:10,140 --> 00:32:15,060
and took part of it
and messed it up.
497
00:32:15,060 --> 00:32:17,420
What we got was
not a vector space.
498
00:32:17,420 --> 00:32:25,670
Now tell me a vector space that
is part of R^2 and is still
499
00:32:25,670 --> 00:32:31,480
safely -- we can multiply, we
can add and we stay in this
500
00:32:31,480 --> 00:32:32,880
smaller vector space.
501
00:32:32,880 --> 00:32:35,680
So it's going to be
called a subspace.
502
00:32:35,680 --> 00:32:40,990
So I'm going to change this
bad example to a good one.
503
00:32:40,990 --> 00:32:42,440
Okay.
504
00:32:42,440 --> 00:32:45,620
So I'm going to
start again with R^2,
505
00:32:45,620 --> 00:32:50,120
but I'm going to take an
example -- it is a vector space,
506
00:32:50,120 --> 00:32:53,805
so it'll be a vector
space inside R^2.
507
00:32:56,970 --> 00:33:03,560
And we'll call that
a subspace of R^2.
508
00:33:06,450 --> 00:33:07,040
Okay.
509
00:33:07,040 --> 00:33:09,010
What can I do?
510
00:33:09,010 --> 00:33:11,960
It's got something in it.
511
00:33:11,960 --> 00:33:14,730
Suppose it's got
this vector in it.
512
00:33:14,730 --> 00:33:17,070
Okay.
513
00:33:17,070 --> 00:33:19,740
If that vector's in
my little subspace
514
00:33:19,740 --> 00:33:23,500
and it's a true
subspace, then there's
515
00:33:23,500 --> 00:33:24,990
got to be some more in it,
516
00:33:24,990 --> 00:33:25,640
right?
517
00:33:25,640 --> 00:33:28,900
I have to be able to
multiply that by two,
518
00:33:28,900 --> 00:33:33,660
and that double vector
has to be included.
519
00:33:33,660 --> 00:33:36,610
Have to be able to multiply
by zero, that vector,
520
00:33:36,610 --> 00:33:39,420
or by half, or by
three quarters.
521
00:33:39,420 --> 00:33:40,310
All these vectors.
522
00:33:40,310 --> 00:33:44,470
Or by minus a half,
or by minus one.
523
00:33:44,470 --> 00:33:48,730
I have to be able to
multiply by any number.
524
00:33:48,730 --> 00:33:52,250
So that is going to say that I
have to have that whole line.
525
00:33:52,250 --> 00:33:56,410
Do you see that?
526
00:33:56,410 --> 00:33:58,440
Once I get a vector in there --
527
00:33:58,440 --> 00:34:03,070
I've got the whole line of
all multiples of that vector.
528
00:34:03,070 --> 00:34:09,320
I can't have a vector space
without extending to get
529
00:34:09,320 --> 00:34:10,770
those multiples in there.
530
00:34:10,770 --> 00:34:12,399
Now I still have
to check addition.
531
00:34:15,000 --> 00:34:16,179
But that comes out okay.
532
00:34:16,179 --> 00:34:20,560
This line is going to work,
because I could add something
533
00:34:20,560 --> 00:34:23,219
on the line to something
else on the line
534
00:34:23,219 --> 00:34:26,540
and I'm still on the line.
535
00:34:26,540 --> 00:34:28,469
So, example.
536
00:34:28,469 --> 00:34:33,340
So this is all examples
of a subspace --
537
00:34:33,340 --> 00:34:45,199
our example is a line in R^2
actually -- not just any line.
538
00:34:45,199 --> 00:34:50,239
If I took this
line, would that --
539
00:34:50,239 --> 00:34:51,960
so all the vectors on that line.
540
00:34:51,960 --> 00:34:56,929
So that vector and that vector
and this vector and this vector
541
00:34:56,929 --> 00:34:58,380
--
542
00:34:58,380 --> 00:35:05,450
in lighter type, I'm drawing
something that doesn't work.
543
00:35:05,450 --> 00:35:07,510
It's not a subspace.
544
00:35:07,510 --> 00:35:09,890
The line in R^2 --
to be a subspace,
545
00:35:09,890 --> 00:35:15,220
the line in R^2 must go
through the zero vector.
546
00:35:19,400 --> 00:35:21,700
Because -- why is
this line no good?
547
00:35:21,700 --> 00:35:23,140
Let me do a dashed line.
548
00:35:27,500 --> 00:35:31,290
Because if I multiplied that
vector on the dashed line
549
00:35:31,290 --> 00:35:34,490
by zero, then I'm down here,
I'm not on the dashed line.
550
00:35:34,490 --> 00:35:36,620
Z- zero's got to be.
551
00:35:36,620 --> 00:35:39,920
Every subspace has
got to contain zero --
552
00:35:39,920 --> 00:35:43,230
because I must be allowed to
multiply by zero and that will
553
00:35:43,230 --> 00:35:46,300
always give me the zero vector.
554
00:35:46,300 --> 00:35:48,020
Okay.
555
00:35:48,020 --> 00:35:51,410
Now, I was going to make --
556
00:35:51,410 --> 00:35:54,460
create some subspaces.
557
00:35:54,460 --> 00:35:59,610
Oh, while I'm in R^2,
why don't we think of all
558
00:35:59,610 --> 00:36:01,110
the possibilities.
559
00:36:01,110 --> 00:36:03,760
R two, there can't be that many.
560
00:36:03,760 --> 00:36:07,510
So what are the possible
subspaces of R^2?
561
00:36:07,510 --> 00:36:08,720
Let me list them.
562
00:36:11,480 --> 00:36:16,220
So I'm listing now
the subspaces of R^2.
563
00:36:19,660 --> 00:36:23,750
And one possibility
that we always allow
564
00:36:23,750 --> 00:36:29,760
is all of R two, the whole
thing, the whole space.
565
00:36:29,760 --> 00:36:34,010
That counts as a
subspace of itself.
566
00:36:34,010 --> 00:36:35,830
You always want to allow that.
567
00:36:35,830 --> 00:36:39,750
Then the others are lines --
568
00:36:39,750 --> 00:36:45,690
any line, meaning infinitely
far in both directions
569
00:36:45,690 --> 00:36:49,810
through the zero.
570
00:36:55,110 --> 00:36:57,790
So that's like
the whole space --
571
00:36:57,790 --> 00:37:00,550
that's like whole two D space.
572
00:37:00,550 --> 00:37:02,860
This is like one dimension.
573
00:37:02,860 --> 00:37:05,320
Is this line the same as R^1 ?
574
00:37:05,320 --> 00:37:07,470
No.
575
00:37:07,470 --> 00:37:11,200
You could say it
looks a lot like R^1.
576
00:37:11,200 --> 00:37:14,380
R^1 was just a line
and this is a line.
577
00:37:14,380 --> 00:37:17,460
But this is a line inside R^2.
578
00:37:17,460 --> 00:37:20,440
The vectors here
have two components.
579
00:37:20,440 --> 00:37:23,600
So that's not the same as R^1,
because there the vectors only
580
00:37:23,600 --> 00:37:25,570
have one component.
581
00:37:25,570 --> 00:37:29,590
Very close, you could
say, but not the same.
582
00:37:29,590 --> 00:37:30,320
Okay.
583
00:37:30,320 --> 00:37:32,250
And now there's a
third possibility.
584
00:37:36,550 --> 00:37:40,940
There's a third
subspace that's --
585
00:37:40,940 --> 00:37:47,970
of R^2 that's not the whole
thing, and it's not a line.
586
00:37:47,970 --> 00:37:50,170
It's even less.
587
00:37:50,170 --> 00:37:52,840
It's just the zero vector alone.
588
00:37:52,840 --> 00:37:55,170
The zero vector alone, only.
589
00:38:01,250 --> 00:38:05,550
I'll often call this
subspace Z, just for zero.
590
00:38:05,550 --> 00:38:07,700
Here's a line, L.
591
00:38:07,700 --> 00:38:10,010
Here's a plane, all of R^2.
592
00:38:10,010 --> 00:38:14,680
So, do you see that
the zero vector's okay?
593
00:38:14,680 --> 00:38:16,970
You would just -- to
understand subspaces,
594
00:38:16,970 --> 00:38:20,820
we have to know the rules -- and
knowing the rules means that we
595
00:38:20,820 --> 00:38:25,040
have to see that yes, the
zero vector by itself,
596
00:38:25,040 --> 00:38:27,990
just this guy alone
satisfies the rules.
597
00:38:27,990 --> 00:38:28,690
Why's that?
598
00:38:28,690 --> 00:38:31,320
Oh, it's too dumb to tell you.
599
00:38:31,320 --> 00:38:36,430
If I took that and added it
to itself, I'm still there.
600
00:38:36,430 --> 00:38:40,320
If I took that and multiplied
by seventeen, I'm still there.
601
00:38:40,320 --> 00:38:44,070
So I've done the operations,
adding and multiplying
602
00:38:44,070 --> 00:38:47,010
by numbers, that are
required, and I didn't go
603
00:38:47,010 --> 00:38:50,300
outside this one point space.
604
00:38:53,570 --> 00:38:57,170
So that's always -- that's
the littlest subspace.
605
00:38:57,170 --> 00:39:00,930
And the largest subspace is the
whole thing and in-between come
606
00:39:00,930 --> 00:39:02,370
all --
607
00:39:02,370 --> 00:39:04,080
whatever's in between.
608
00:39:04,080 --> 00:39:04,580
Okay.
609
00:39:04,580 --> 00:39:07,610
So for example, what's
in between for R^3?
610
00:39:07,610 --> 00:39:12,100
So if I'm in ordinary three
dimensions, the subspace is R,
611
00:39:12,100 --> 00:39:18,250
all of R^3 at one extreme,
the zero vector at the bottom.
612
00:39:18,250 --> 00:39:23,430
And then a plane, a
plane through the origin.
613
00:39:23,430 --> 00:39:26,510
Or a line, a line
through the origin.
614
00:39:26,510 --> 00:39:32,970
So with R^3, the subspaces were
R^3, plane through the origin,
615
00:39:32,970 --> 00:39:37,560
line through the origin and
a zero vector by itself,
616
00:39:37,560 --> 00:39:43,030
zero zero zero, just
that single vector.
617
00:39:43,030 --> 00:39:44,360
Okay, you've got the idea.
618
00:39:47,470 --> 00:39:51,080
But, now comes --
619
00:39:51,080 --> 00:39:53,350
the reality is --
620
00:39:53,350 --> 00:39:57,530
what are these -- where
do these subspaces come --
621
00:39:57,530 --> 00:40:00,950
how do they come
out of matrices?
622
00:40:00,950 --> 00:40:06,080
And I want to take
this matrix --
623
00:40:06,080 --> 00:40:08,350
oh, let me take that matrix.
624
00:40:08,350 --> 00:40:17,430
So I want to create some
subspaces out of that matrix.
625
00:40:17,430 --> 00:40:26,980
Well, one subspace
is from the columns.
626
00:40:26,980 --> 00:40:29,760
Okay.
627
00:40:29,760 --> 00:40:34,050
So this is the
important subspace,
628
00:40:34,050 --> 00:40:38,190
the first important subspace
that comes from that matrix --
629
00:40:38,190 --> 00:40:40,750
I'm going to -- let
me call it A again.
630
00:40:40,750 --> 00:40:44,370
Back to -- okay.
631
00:40:44,370 --> 00:40:48,150
I'm looking at the columns of A.
632
00:40:48,150 --> 00:40:50,530
Those are vectors in R^3.
633
00:40:50,530 --> 00:40:52,380
So the columns are in R^3.
634
00:40:52,380 --> 00:40:58,100
The columns are in R^3.
635
00:41:02,280 --> 00:41:04,585
So I want those columns
to be in my subspace.
636
00:41:08,970 --> 00:41:11,960
Now I can't just put two
columns in my subspace
637
00:41:11,960 --> 00:41:14,512
and call it a subspace.
638
00:41:14,512 --> 00:41:16,970
What do I have to throw in --
if I'm going to put those two
639
00:41:16,970 --> 00:41:21,460
columns in, what else has got
to be there to have a subspace?
640
00:41:21,460 --> 00:41:25,050
I must be able to
add those things.
641
00:41:25,050 --> 00:41:28,460
So the sum of those columns --
642
00:41:28,460 --> 00:41:34,970
so these columns are in R^3,
and I have to be able --
643
00:41:34,970 --> 00:41:37,330
I'm, you know, I want
that to be in my subspace,
644
00:41:37,330 --> 00:41:39,080
I want that to be
in my subspace,
645
00:41:39,080 --> 00:41:42,880
but therefore I have to be able
to multiply them by anything.
646
00:41:42,880 --> 00:41:45,910
Zero zero zero has got
to be in my subspace.
647
00:41:45,910 --> 00:41:48,630
I have to be able to add
them so that four five five
648
00:41:48,630 --> 00:41:50,150
is in the subspace.
649
00:41:50,150 --> 00:41:53,054
I've got to be able to add one
of these plus three of these.
650
00:41:53,054 --> 00:41:54,470
That'll give me
some other vector.
651
00:41:57,100 --> 00:42:02,180
I have to be able to take
all the linear combinations.
652
00:42:02,180 --> 00:42:14,200
So these are columns in R^3 and
all there linear combinations
653
00:42:14,200 --> 00:42:16,920
form a subspace.
654
00:42:21,260 --> 00:42:23,400
What do I mean by
linear combinations?
655
00:42:23,400 --> 00:42:26,060
I mean multiply
that by something,
656
00:42:26,060 --> 00:42:28,290
multiply that by
something and add.
657
00:42:28,290 --> 00:42:33,350
The two operations of linear
algebra, multiplying by numbers
658
00:42:33,350 --> 00:42:36,060
and adding vectors.
659
00:42:36,060 --> 00:42:38,930
And, if I include
all the results,
660
00:42:38,930 --> 00:42:40,875
then I'm guaranteed
to have a subspace.
661
00:42:43,570 --> 00:42:46,860
I've done the job.
662
00:42:46,860 --> 00:42:49,210
And we'll give it a name --
663
00:42:49,210 --> 00:42:49,960
the column space.
664
00:42:53,740 --> 00:42:54,380
Column space.
665
00:43:01,220 --> 00:43:05,920
And maybe I'll call it C of A.
666
00:43:05,920 --> 00:43:07,120
C for column space.
667
00:43:11,580 --> 00:43:15,020
There's an idea there that --
668
00:43:15,020 --> 00:43:22,750
Like, the central idea
for today's lecture is --
669
00:43:22,750 --> 00:43:25,220
got a few vectors.
670
00:43:25,220 --> 00:43:27,130
Not satisfied with
a few vectors,
671
00:43:27,130 --> 00:43:29,800
we want a space of vectors.
672
00:43:29,800 --> 00:43:33,160
The vectors, they're in --
these vectors in -- are in R^3 ,
673
00:43:33,160 --> 00:43:37,050
so our space of vectors
will be vectors in R^3.
674
00:43:37,050 --> 00:43:40,940
The key idea's -- we
have to be able to take
675
00:43:40,940 --> 00:43:42,400
their combinations.
676
00:43:42,400 --> 00:43:47,300
So tell me, geometrically,
if I drew all these things --
677
00:43:47,300 --> 00:43:50,060
like if I drew one two four,
that would be somewhere maybe
678
00:43:50,060 --> 00:43:50,930
there.
679
00:43:50,930 --> 00:43:54,740
If I drew three three one,
who knows, might be --
680
00:43:54,740 --> 00:43:57,140
I don't know, I'll say there.
681
00:43:57,140 --> 00:44:01,690
There's column one,
there's column two.
682
00:44:01,690 --> 00:44:06,700
What else -- what's in
the whole column space?
683
00:44:06,700 --> 00:44:11,160
How do I draw the
whole column space now?
684
00:44:11,160 --> 00:44:13,430
I take all combinations
of those two vectors.
685
00:44:15,970 --> 00:44:18,220
Do I get -- well, I
guess I actually listed
686
00:44:18,220 --> 00:44:19,160
the possibilities.
687
00:44:19,160 --> 00:44:21,940
Do I get the whole space?
688
00:44:21,940 --> 00:44:24,190
Do I get a plane?
689
00:44:24,190 --> 00:44:26,984
I get more than a
line, that's for sure.
690
00:44:26,984 --> 00:44:28,900
And I certainly get more
than the zero vector,
691
00:44:28,900 --> 00:44:31,610
but I do get the
zero vector included.
692
00:44:31,610 --> 00:44:34,160
What do I get if I combine --
693
00:44:34,160 --> 00:44:39,115
take all the combinations
of two vectors in R^3 ?
694
00:44:44,040 --> 00:44:46,450
So I've got all this stuff on --
695
00:44:46,450 --> 00:44:49,040
that whole line gets filled
out, that whole line gets filled
696
00:44:49,040 --> 00:44:51,190
out, but all in-between
gets filled out --
697
00:44:51,190 --> 00:44:52,900
between the two
lines because I --
698
00:44:52,900 --> 00:44:56,610
I allowed to add something
from one line, something
699
00:44:56,610 --> 00:44:57,850
from the other.
700
00:44:57,850 --> 00:44:58,810
You see what's coming?
701
00:44:58,810 --> 00:44:59,643
I'm getting a plane.
702
00:45:05,060 --> 00:45:06,790
That's my -- and it's
through the origin.
703
00:45:10,210 --> 00:45:17,950
Those two vectors, namely one
two four and three three one,
704
00:45:17,950 --> 00:45:20,590
when I take all
their combinations,
705
00:45:20,590 --> 00:45:21,770
I fill out a whole plane.
706
00:45:21,770 --> 00:45:25,240
Please think about that.
707
00:45:25,240 --> 00:45:28,280
That's the picture
you have to see.
708
00:45:28,280 --> 00:45:31,940
You sure have to see it in R^3
, because we're going to do it
709
00:45:31,940 --> 00:45:36,880
in R^10, and we may take a
combination of five vectors
710
00:45:36,880 --> 00:45:40,740
in R^10, and what will we have?
711
00:45:40,740 --> 00:45:41,630
God knows.
712
00:45:41,630 --> 00:45:44,910
It's some subspace.
713
00:45:44,910 --> 00:45:46,880
We'll have five vectors.
714
00:45:46,880 --> 00:45:49,010
They'll all have ten components.
715
00:45:49,010 --> 00:45:52,320
We take their combinations.
716
00:45:52,320 --> 00:45:58,240
We don't have R^5 , because our
vectors have ten components.
717
00:45:58,240 --> 00:46:05,020
And we possibly have, like,
some five dimensional flat thing
718
00:46:05,020 --> 00:46:06,680
going through the
origin for sure.
719
00:46:09,220 --> 00:46:12,110
Well, of course, if those five
vectors were all on the line,
720
00:46:12,110 --> 00:46:13,710
then we would only
get that line.
721
00:46:13,710 --> 00:46:16,840
So, you see, there are, like,
other possibilities here.
722
00:46:16,840 --> 00:46:21,690
It depends what -- it depends
on those five vectors.
723
00:46:21,690 --> 00:46:25,440
Just like if our two columns
had been on the same line,
724
00:46:25,440 --> 00:46:28,640
then the column space would
have been only a line.
725
00:46:28,640 --> 00:46:31,440
Here it was a plane.
726
00:46:31,440 --> 00:46:31,940
Okay.
727
00:46:35,700 --> 00:46:37,610
I'm going to stop at that point.
728
00:46:37,610 --> 00:46:44,220
That's the central idea of
-- the great example of how
729
00:46:44,220 --> 00:46:48,960
to create a subspace
from a matrix.
730
00:46:48,960 --> 00:46:51,990
Take its columns, take
their combinations,
731
00:46:51,990 --> 00:46:57,360
all their linear combinations
and you get the column space.
732
00:46:57,360 --> 00:47:01,060
And that's the
central sort of --
733
00:47:01,060 --> 00:47:04,320
we're looking at linear
algebra at a higher level.
734
00:47:04,320 --> 00:47:07,600
When I look at A -- now,
I want to look at Ax=b.
735
00:47:07,600 --> 00:47:10,090
That'll be the first
thing in the next lecture.
736
00:47:13,650 --> 00:47:17,300
How do I understand
Ax=b in this language --
737
00:47:17,300 --> 00:47:22,580
in this new language of vector
spaces and column spaces.
738
00:47:22,580 --> 00:47:24,830
And what are other subspaces?
739
00:47:24,830 --> 00:47:30,230
So the column space is a big
one, there are others to come.
740
00:47:30,230 --> 00:47:32,270
Okay, thanks.