WEBVTT
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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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And they're easy to
identify, of course.
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You know, it's not maybe so
easy before we had a case where
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the transpose gave the inverse.
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That's highly important,
but not so simple to see.
00:14:36.230 --> 00:14:39.200
This is the case where the
transpose gives the same matrix
00:14:39.200 --> 00:14:39.720
back again.
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That's totally simple to see.
00:14:42.510 --> 00:14:43.380
Okay.
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Could actually -- maybe I could
even say when would we get such
00:14:49.410 --> 00:14:51.150
a matrix?
00:14:51.150 --> 00:14:55.740
For example, this -- that
matrix is absolutely far from
00:14:55.740 --> 00:14:57.850
symmetric, right?
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 --
00:15:03.700 --> 00:15:06.580
lies down on its side.
00:15:06.580 --> 00:15:10.970
But let me tell you a way to
get a symmetric matrix out of
00:15:10.970 --> 00:15:11.880
this.
00:15:11.880 --> 00:15:14.640
Multiply those together.
00:15:14.640 --> 00:15:17.210
If I multiply this
rectangular, shall I
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call it R for rectangular?
00:15:18.910 --> 00:15:22.890
So let that be R for
rectangular matrix
00:15:22.890 --> 00:15:27.580
and let that be R
transpose, which it is.
00:15:27.580 --> 00:15:32.000
Then I think that if I
multiply those together,
00:15:32.000 --> 00:15:34.390
I get a symmetric matrix.
00:15:34.390 --> 00:15:36.780
Can I just do it
with the numbers
00:15:36.780 --> 00:15:44.700
and then ask you why, how did
I know it would be symmetric?
00:15:44.700 --> 00:15:52.415
So my point is that R transpose
R is always symmetric.
00:15:52.415 --> 00:15:52.914
Okay?
00:15:57.560 --> 00:16:01.470
And I'm going to do it for that
particular R transpose R which
00:16:01.470 --> 00:16:02.750
was --
00:16:02.750 --> 00:16:05.815
let's see, the column was
one two four three three one.
00:16:09.700 --> 00:16:11.700
I called that one R
transpose, didn't I,
00:16:11.700 --> 00:16:16.030
and I called this guy one
two four three three one.
00:16:16.030 --> 00:16:17.550
I called that R.
00:16:17.550 --> 00:16:19.970
Shall we just do
that multiplication?
00:16:19.970 --> 00:16:22.960
Okay, so up here
I'm getting a ten.
00:16:22.960 --> 00:16:27.310
Next to it I'm getting two, a
nine, I'm getting an eleven.
00:16:27.310 --> 00:16:30.390
Next to that I'm getting
four and three, a seven.
00:16:30.390 --> 00:16:32.650
Now what do I get there?
00:16:32.650 --> 00:16:37.650
This eleven came from one
three times two three, right?
00:16:37.650 --> 00:16:39.720
Row one, column two.
00:16:39.720 --> 00:16:41.530
What goes here?
00:16:41.530 --> 00:16:43.360
Row two, column one.
00:16:43.360 --> 00:16:46.050
But no difference.
00:16:46.050 --> 00:16:50.220
One three two three or two
three one three, same thing.
00:16:50.220 --> 00:16:51.750
It's going to be an eleven.
00:16:51.750 --> 00:16:54.310
That's the symmetry.
00:16:54.310 --> 00:16:56.230
I can continue to fill it out.
00:16:56.230 --> 00:16:58.100
What -- oh, let's
get that seven.
00:16:58.100 --> 00:17:00.520
That seven will show
up down here, too,
00:17:00.520 --> 00:17:02.650
and then four more numbers.
00:17:02.650 --> 00:17:06.619
That seven will show up here
because one three times four
00:17:06.619 --> 00:17:10.630
one gave the seven, but also
four one times one three
00:17:10.630 --> 00:17:11.599
will give that seven.
00:17:11.599 --> 00:17:16.030
Do you see that it works?
00:17:16.030 --> 00:17:24.630
Actually, do you want to see it
work also in matrix language?
00:17:24.630 --> 00:17:27.089
I mean, that's quite
convincing, right?
00:17:27.089 --> 00:17:29.965
That seven is no accident.
00:17:32.630 --> 00:17:35.550
The eleven is no accident.
00:17:35.550 --> 00:17:40.680
But just tell me how do I know
if I transpose this guy --
00:17:40.680 --> 00:17:42.820
How do I know it's symmetric?
00:17:42.820 --> 00:17:45.620
Well, I'm going to transpose it.
00:17:45.620 --> 00:17:49.610
And when I transpose
it, I'm hoping
00:17:49.610 --> 00:17:52.210
I get the matrix back again.
00:17:52.210 --> 00:17:54.630
So can I transpose
R transpose R?
00:17:54.630 --> 00:17:56.157
So just -- so, why?
00:17:59.880 --> 00:18:02.945
Well, my suggestion
is take the transpose.
00:18:09.800 --> 00:18:11.670
That's the only way to
show it's symmetric.
00:18:11.670 --> 00:18:14.340
Take the transpose and
see that it didn't change.
00:18:14.340 --> 00:18:19.970
Okay, so I take the
transpose of R transpose R.
00:18:19.970 --> 00:18:20.470
Okay.
00:18:20.470 --> 00:18:23.190
How do I do that?
00:18:23.190 --> 00:18:27.680
This is our little practice
on the rules for transposes.
00:18:27.680 --> 00:18:33.780
So the rule for transposes
is the order gets reversed.
00:18:33.780 --> 00:18:38.300
Just like inverses,
which we did prove,
00:18:38.300 --> 00:18:44.390
same rule for transposes
and -- which we'll now use.
00:18:44.390 --> 00:18:46.110
So the order gets reversed.
00:18:46.110 --> 00:18:51.600
It's the transpose of
that that comes first,
00:18:51.600 --> 00:18:56.880
and the transpose of
this that comes -- no.
00:18:56.880 --> 00:18:58.010
Is that -- yeah.
00:18:58.010 --> 00:19:01.130
That's what I have
to write, right?
00:19:01.130 --> 00:19:03.500
This is a product of two
matrices and I want its
00:19:03.500 --> 00:19:04.350
transpose.
00:19:04.350 --> 00:19:06.880
So I put the matrices
in the opposite order
00:19:06.880 --> 00:19:08.480
and I transpose them.
00:19:08.480 --> 00:19:09.780
But what have I got here?
00:19:09.780 --> 00:19:12.320
What is R transpose transpose?
00:19:15.300 --> 00:19:18.670
Well, don't all speak at once.
00:19:18.670 --> 00:19:22.080
R transpose transpose, I
flipped over the diagonal,
00:19:22.080 --> 00:19:28.540
I flipped over the diagonal
again, so I've got R.
00:19:28.540 --> 00:19:32.770
And that's just my point, that
if I started with this matrix,
00:19:32.770 --> 00:19:34.740
I transposed it, I
got it back again.
00:19:37.850 --> 00:19:46.810
So that's the check, without
using numbers, but with --
00:19:46.810 --> 00:19:52.510
it checked in two lines that I
always get symmetric matrices
00:19:52.510 --> 00:19:53.480
this way.
00:19:53.480 --> 00:19:55.680
And actually, that's
where they come
00:19:55.680 --> 00:20:00.220
from in so many
practical applications.
00:20:00.220 --> 00:20:02.730
Okay.
00:20:02.730 --> 00:20:07.580
So now I've said something today
about permutations and about
00:20:07.580 --> 00:20:14.540
transposes and about
symmetry and I'm ready
00:20:14.540 --> 00:20:17.290
for chapter three.
00:20:17.290 --> 00:20:21.550
Can we take a breath --
00:20:21.550 --> 00:20:25.250
the tape won't take a breath,
but the lecturer will,
00:20:25.250 --> 00:20:31.980
because to tell you
about vector spaces is --
00:20:31.980 --> 00:20:38.220
we really have to start now
and think, okay, listen up.
00:20:38.220 --> 00:20:39.500
What are vector spaces?
00:20:47.400 --> 00:20:48.400
And what are sub-spaces?
00:20:51.160 --> 00:20:51.890
Okay.
00:20:51.890 --> 00:21:01.430
So, the point is, The main
operations that we do --
00:21:01.430 --> 00:21:04.450
what do we do with vectors?
00:21:04.450 --> 00:21:05.820
We add them.
00:21:05.820 --> 00:21:07.270
We know how to add two vectors.
00:21:09.880 --> 00:21:13.930
We multiply them by numbers,
usually called scalers.
00:21:13.930 --> 00:21:17.350
If we have a vector, we
know what three V is.
00:21:17.350 --> 00:21:24.030
If we have a vector V and
W, we know what V plus W is.
00:21:24.030 --> 00:21:26.670
Those are the two
operations that we've
00:21:26.670 --> 00:21:29.090
got to be able to do.
00:21:29.090 --> 00:21:33.220
To legitimately talk
about a space of vectors,
00:21:33.220 --> 00:21:35.870
the requirement
is that we should
00:21:35.870 --> 00:21:39.990
be able to add the things
and multiply by numbers
00:21:39.990 --> 00:21:44.380
and that there should be
some decent rules satisfied.
00:21:44.380 --> 00:21:45.370
Okay.
00:21:45.370 --> 00:21:48.590
So let me start with examples.
00:21:48.590 --> 00:21:50.755
So I'm talking now
about vector spaces.
00:21:56.810 --> 00:21:58.685
And I'm going to
start with examples.
00:22:06.690 --> 00:22:09.480
Let me say again what this
word space is meaning.
00:22:09.480 --> 00:22:14.010
When I say that word
space, that means to me
00:22:14.010 --> 00:22:19.440
that I've got a bunch of
vectors, a space of vectors.
00:22:19.440 --> 00:22:21.010
But not just any
bunch of vectors.
00:22:24.010 --> 00:22:28.720
It has to be a
space of vectors --
00:22:28.720 --> 00:22:31.300
has to allow me to do the
operations that vectors
00:22:31.300 --> 00:22:32.740
are for.
00:22:32.740 --> 00:22:37.110
I have to be able to add
vectors and multiply by numbers.
00:22:37.110 --> 00:22:39.560
I have to be able to
take linear combinations.
00:22:39.560 --> 00:22:43.120
Well, where did we meet
linear combinations?
00:22:43.120 --> 00:22:48.960
We met them back in, say in R^2.
00:22:48.960 --> 00:22:51.370
So there's a vector space.
00:22:51.370 --> 00:22:54.010
What's that vector space?
00:22:54.010 --> 00:22:59.270
So R two is telling me I'm
talking about real numbers
00:22:59.270 --> 00:23:01.370
and I'm talking about
two real numbers.
00:23:01.370 --> 00:23:11.470
So this is all two
dimensional vectors --
00:23:11.470 --> 00:23:16.580
real, such as --
00:23:16.580 --> 00:23:18.750
well, I'm not going to
be able to list them all.
00:23:18.750 --> 00:23:20.210
But let me put a few down.
00:23:20.210 --> 00:23:30.420
|3; 2|, |0;0|, |pi; e|.
00:23:30.420 --> 00:23:30.920
So on.
00:23:35.450 --> 00:23:39.890
And it's natural -- okay.
00:23:39.890 --> 00:23:44.270
Let's see, I guess I
should do algebra first.
00:23:44.270 --> 00:23:46.980
Algebra means what can
I do to these vectors?
00:23:46.980 --> 00:23:48.130
I can add them.
00:23:48.130 --> 00:23:50.520
I can add that to that.
00:23:50.520 --> 00:23:51.660
And how do I do it?
00:23:51.660 --> 00:23:54.460
A component at a
time, of course.
00:23:54.460 --> 00:23:58.240
Three two added to zero
zero gives me, three two.
00:23:58.240 --> 00:24:00.190
Sorry about that.
00:24:00.190 --> 00:24:05.780
Three two added to pi e gives
me three plus pi, two plus e.
00:24:05.780 --> 00:24:07.260
Oh, you know what it does.
00:24:07.260 --> 00:24:11.240
And you know the picture
that goes with it.
00:24:11.240 --> 00:24:14.830
There's the vector three two.
00:24:14.830 --> 00:24:19.520
And often, the
picture has an arrow.
00:24:19.520 --> 00:24:22.550
The vector zero zero, which is
a highly important vector --
00:24:22.550 --> 00:24:24.610
it's got, like, the
most important here
00:24:24.610 --> 00:24:25.930
-- is there.
00:24:25.930 --> 00:24:29.840
And of course there's
not much of an arrow.
00:24:29.840 --> 00:24:35.110
Pi -- I'll have to remember --
pi is about three and a little
00:24:35.110 --> 00:24:37.570
more, e is about two
and a little more.
00:24:37.570 --> 00:24:41.090
So maybe there's pi e.
00:24:41.090 --> 00:24:44.690
I never drew pi e before.
00:24:44.690 --> 00:24:47.030
It's just natural to --
00:24:47.030 --> 00:24:55.560
this is the first
component on the horizontal
00:24:55.560 --> 00:24:59.470
and this is the
second component,
00:24:59.470 --> 00:25:02.010
going up the vertical.
00:25:02.010 --> 00:25:02.910
Okay.
00:25:02.910 --> 00:25:07.570
And the whole plane is R two.
00:25:07.570 --> 00:25:14.980
So R two is, we
could say, the plane.
00:25:14.980 --> 00:25:17.710
The xy plane.
00:25:17.710 --> 00:25:18.920
That's what everybody thinks.
00:25:24.770 --> 00:25:32.800
But the point is it's a vector
space because all those vectors
00:25:32.800 --> 00:25:34.140
are in there.
00:25:34.140 --> 00:25:37.380
If I removed one of them --
00:25:37.380 --> 00:25:39.420
Suppose I removed zero zero.
00:25:39.420 --> 00:25:43.480
Suppose I tried to take the --
considered the X Y plane with
00:25:43.480 --> 00:25:46.360
a puncture, with
a point removed.
00:25:46.360 --> 00:25:47.200
Like the origin.
00:25:47.200 --> 00:25:50.470
That would be, like, awful
to take the origin away.
00:25:50.470 --> 00:25:52.570
Why is that?
00:25:52.570 --> 00:25:54.550
Why do I need the origin there?
00:25:54.550 --> 00:25:59.570
Because I have to be allowed --
if I had these other vectors,
00:25:59.570 --> 00:26:03.330
I have to be allowed to
multiply three two --
00:26:03.330 --> 00:26:05.610
this was three two --
00:26:05.610 --> 00:26:09.820
by anything, by any
scaler, including zero.
00:26:09.820 --> 00:26:12.020
I've got to be allowed
to multiply by zero
00:26:12.020 --> 00:26:15.010
and the result's
got to be there.
00:26:15.010 --> 00:26:18.110
I can't do without that point.
00:26:18.110 --> 00:26:23.670
And I have to be able to add
three two to the opposite guy,
00:26:23.670 --> 00:26:26.800
minus three minus two.
00:26:26.800 --> 00:26:29.230
And if I add those I'm
back to the origin again.
00:26:29.230 --> 00:26:31.360
No way I can do
without the origin.
00:26:31.360 --> 00:26:36.280
Every vector space has got
that zero vector in it.
00:26:36.280 --> 00:26:38.650
Okay, that's an
easy vector space,
00:26:38.650 --> 00:26:42.520
because we have a
natural picture of it.
00:26:42.520 --> 00:26:43.840
Okay.
00:26:43.840 --> 00:26:46.260
Similarly easy is R^3.
00:26:50.340 --> 00:26:54.630
This would be all -- let
me go up a little here.
00:26:54.630 --> 00:26:57.820
This would be --
00:26:57.820 --> 00:27:02.670
R three would be all three
dimensional vectors --
00:27:02.670 --> 00:27:09.645
or shall I say vectors
with three real components.
00:27:14.320 --> 00:27:15.000
Okay.
00:27:15.000 --> 00:27:21.030
Let me just to be
sure we're together,
00:27:21.030 --> 00:27:23.661
let me take the
vector three two zero.
00:27:29.410 --> 00:27:33.390
Is that a vector in R^2 or R^3?
00:27:33.390 --> 00:27:38.490
Definitely it's in R^3.
00:27:38.490 --> 00:27:40.150
It's got three components.
00:27:40.150 --> 00:27:43.040
One of them happens to be zero,
but that's a perfectly okay
00:27:43.040 --> 00:27:43.850
number.
00:27:43.850 --> 00:27:48.290
So that's a vector in R^3.
00:27:48.290 --> 00:27:51.590
We don't want to mix up the --
00:27:51.590 --> 00:27:55.090
I mean, keep these vectors
straight and keep R^n straight.
00:27:55.090 --> 00:27:57.630
So what's R^n?
00:27:57.630 --> 00:27:59.150
R^n.
00:27:59.150 --> 00:28:05.990
So this is our big example, is
all vectors with n components.
00:28:05.990 --> 00:28:11.170
And I'm making these darn
things column vectors.
00:28:11.170 --> 00:28:14.050
Can I try to follow
that convention,
00:28:14.050 --> 00:28:17.530
that they'll be column vectors,
and their components should
00:28:17.530 --> 00:28:20.690
be real numbers.
00:28:20.690 --> 00:28:24.830
Later we'll need complex
numbers and complex vectors,
00:28:24.830 --> 00:28:26.721
but much later.
00:28:26.721 --> 00:28:27.220
Okay.
00:28:27.220 --> 00:28:28.780
So that's a vector space.
00:28:31.420 --> 00:28:33.670
Now, let's see.
00:28:33.670 --> 00:28:35.910
What do I have to tell
you about vector spaces?
00:28:35.910 --> 00:28:44.090
I said the most important thing,
which is that we can add any
00:28:44.090 --> 00:28:46.760
two of these and
we -- still in R^2.
00:28:46.760 --> 00:28:50.220
We can multiply by any number
and we're still in R^2.
00:28:50.220 --> 00:28:53.380
We can take any combination
and we're still in R^2.
00:28:53.380 --> 00:28:55.290
And same goes for R^n.
00:28:55.290 --> 00:29:02.240
It's -- honesty requires me to
mention that these operations
00:29:02.240 --> 00:29:08.300
of adding and multiplying
have to obey a few rules.
00:29:08.300 --> 00:29:12.790
Like, we can't just arbitrarily
say, okay, the sum of three two
00:29:12.790 --> 00:29:15.610
and pi e is zero zero.
00:29:15.610 --> 00:29:18.410
It's not.
00:29:18.410 --> 00:29:22.650
The sum of three two and
minus three two is zero zero.
00:29:22.650 --> 00:29:27.030
So -- oh, I'm not going
to -- the book, actually,
00:29:27.030 --> 00:29:32.420
lists the eight rules that the
addition and multiplication
00:29:32.420 --> 00:29:34.680
have to satisfy, but they do.
00:29:34.680 --> 00:29:38.810
They certainly satisfy it in
R^n and usually it's not those
00:29:38.810 --> 00:29:42.170
eight rules that are in doubt.
00:29:42.170 --> 00:29:50.070
What's -- the question is, can
we do those additions and do we
00:29:50.070 --> 00:29:51.250
stay in the space?
00:29:51.250 --> 00:29:55.580
Let me show you a
case where you can't.
00:29:55.580 --> 00:29:59.810
So suppose this is going
to be not a vector space.
00:30:05.490 --> 00:30:08.780
Suppose I take the xy
plane -- so there's R^2.
00:30:08.780 --> 00:30:11.240
That is a vector space.
00:30:11.240 --> 00:30:15.940
Now suppose I just
take part of it.
00:30:15.940 --> 00:30:17.670
Just this.
00:30:17.670 --> 00:30:22.270
Just this one -- this is one
quarter of the vector space.
00:30:24.910 --> 00:30:29.965
All the vectors with positive
or at least not negative
00:30:29.965 --> 00:30:30.465
components.
00:30:33.070 --> 00:30:37.540
Can I add those safely?
00:30:37.540 --> 00:30:38.410
Yes.
00:30:38.410 --> 00:30:41.690
If I add a vector
with, like, two --
00:30:41.690 --> 00:30:45.030
three two to another
vector like five six,
00:30:45.030 --> 00:30:48.950
I'm still up in this quarter,
no problem with adding.
00:30:48.950 --> 00:30:54.860
But there's a heck of a problem
with multiplying by scalers,
00:30:54.860 --> 00:30:58.690
because there's a lot of
scalers that will take me out
00:30:58.690 --> 00:31:02.280
of this quarter plane,
like negative ones.
00:31:02.280 --> 00:31:05.820
If I took three two and I
multiplied by minus five,
00:31:05.820 --> 00:31:08.240
I'm way down here.
00:31:08.240 --> 00:31:12.220
So that's not a vector
space, because it's not --
00:31:12.220 --> 00:31:14.250
closed is the right word.
00:31:14.250 --> 00:31:17.870
It's not closed
under multiplication
00:31:17.870 --> 00:31:19.850
by all real numbers.
00:31:22.500 --> 00:31:27.150
So a vector space has to be
closed under multiplication
00:31:27.150 --> 00:31:29.010
and addition of vectors.
00:31:29.010 --> 00:31:31.680
In other words,
linear combinations.
00:31:31.680 --> 00:31:37.560
It -- so, it means that if
I give you a few vectors --
00:31:37.560 --> 00:31:39.980
yeah look, here's an
important -- here --
00:31:39.980 --> 00:31:42.420
now we're getting to some
really important vector spaces.
00:31:42.420 --> 00:31:47.460
Well, R^n -- like, they
are the most important.
00:31:47.460 --> 00:31:52.520
But we will be interested in
so- in vector spaces that are
00:31:52.520 --> 00:31:55.700
inside R^n.
00:31:55.700 --> 00:32:01.790
Vector spaces that follow
the rules, but they --
00:32:01.790 --> 00:32:10.140
we don't need all of -- see,
there we started with R^2 here,
00:32:10.140 --> 00:32:15.060
and took part of it
and messed it up.
00:32:15.060 --> 00:32:17.420
What we got was
not a vector space.
00:32:17.420 --> 00:32:25.670
Now tell me a vector space that
is part of R^2 and is still
00:32:25.670 --> 00:32:31.480
safely -- we can multiply, we
can add and we stay in this
00:32:31.480 --> 00:32:32.880
smaller vector space.
00:32:32.880 --> 00:32:35.680
So it's going to be
called a subspace.
00:32:35.680 --> 00:32:40.990
So I'm going to change this
bad example to a good one.
00:32:40.990 --> 00:32:42.440
Okay.
00:32:42.440 --> 00:32:45.620
So I'm going to
start again with R^2,
00:32:45.620 --> 00:32:50.120
but I'm going to take an
example -- it is a vector space,
00:32:50.120 --> 00:32:53.805
so it'll be a vector
space inside R^2.
00:32:56.970 --> 00:33:03.560
And we'll call that
a subspace of R^2.
00:33:06.450 --> 00:33:07.040
Okay.
00:33:07.040 --> 00:33:09.010
What can I do?
00:33:09.010 --> 00:33:11.960
It's got something in it.
00:33:11.960 --> 00:33:14.730
Suppose it's got
this vector in it.
00:33:14.730 --> 00:33:17.070
Okay.
00:33:17.070 --> 00:33:19.740
If that vector's in
my little subspace
00:33:19.740 --> 00:33:23.500
and it's a true
subspace, then there's
00:33:23.500 --> 00:33:24.990
got to be some more in it,
00:33:24.990 --> 00:33:25.640
right?
00:33:25.640 --> 00:33:28.900
I have to be able to
multiply that by two,
00:33:28.900 --> 00:33:33.660
and that double vector
has to be included.
00:33:33.660 --> 00:33:36.610
Have to be able to multiply
by zero, that vector,
00:33:36.610 --> 00:33:39.420
or by half, or by
three quarters.
00:33:39.420 --> 00:33:40.310
All these vectors.
00:33:40.310 --> 00:33:44.470
Or by minus a half,
or by minus one.
00:33:44.470 --> 00:33:48.730
I have to be able to
multiply by any number.
00:33:48.730 --> 00:33:52.250
So that is going to say that I
have to have that whole line.
00:33:52.250 --> 00:33:56.410
Do you see that?
00:33:56.410 --> 00:33:58.440
Once I get a vector in there --
00:33:58.440 --> 00:34:03.070
I've got the whole line of
all multiples of that vector.
00:34:03.070 --> 00:34:09.320
I can't have a vector space
without extending to get
00:34:09.320 --> 00:34:10.770
those multiples in there.
00:34:10.770 --> 00:34:12.399
Now I still have
to check addition.
00:34:15.000 --> 00:34:16.179
But that comes out okay.
00:34:16.179 --> 00:34:20.560
This line is going to work,
because I could add something
00:34:20.560 --> 00:34:23.219
on the line to something
else on the line
00:34:23.219 --> 00:34:26.540
and I'm still on the line.
00:34:26.540 --> 00:34:28.469
So, example.
00:34:28.469 --> 00:34:33.340
So this is all examples
of a subspace --
00:34:33.340 --> 00:34:45.199
our example is a line in R^2
actually -- not just any line.
00:34:45.199 --> 00:34:50.239
If I took this
line, would that --
00:34:50.239 --> 00:34:51.960
so all the vectors on that line.
00:34:51.960 --> 00:34:56.929
So that vector and that vector
and this vector and this vector
00:34:56.929 --> 00:34:58.380
--
00:34:58.380 --> 00:35:05.450
in lighter type, I'm drawing
something that doesn't work.
00:35:05.450 --> 00:35:07.510
It's not a subspace.
00:35:07.510 --> 00:35:09.890
The line in R^2 --
to be a subspace,
00:35:09.890 --> 00:35:15.220
the line in R^2 must go
through the zero vector.
00:35:19.400 --> 00:35:21.700
Because -- why is
this line no good?
00:35:21.700 --> 00:35:23.140
Let me do a dashed line.
00:35:27.500 --> 00:35:31.290
Because if I multiplied that
vector on the dashed line
00:35:31.290 --> 00:35:34.490
by zero, then I'm down here,
I'm not on the dashed line.
00:35:34.490 --> 00:35:36.620
Z- zero's got to be.
00:35:36.620 --> 00:35:39.920
Every subspace has
got to contain zero --
00:35:39.920 --> 00:35:43.230
because I must be allowed to
multiply by zero and that will
00:35:43.230 --> 00:35:46.300
always give me the zero vector.
00:35:46.300 --> 00:35:48.020
Okay.
00:35:48.020 --> 00:35:51.410
Now, I was going to make --
00:35:51.410 --> 00:35:54.460
create some subspaces.
00:35:54.460 --> 00:35:59.610
Oh, while I'm in R^2,
why don't we think of all
00:35:59.610 --> 00:36:01.110
the possibilities.
00:36:01.110 --> 00:36:03.760
R two, there can't be that many.
00:36:03.760 --> 00:36:07.510
So what are the possible
subspaces of R^2?
00:36:07.510 --> 00:36:08.720
Let me list them.
00:36:11.480 --> 00:36:16.220
So I'm listing now
the subspaces of R^2.
00:36:19.660 --> 00:36:23.750
And one possibility
that we always allow
00:36:23.750 --> 00:36:29.760
is all of R two, the whole
thing, the whole space.
00:36:29.760 --> 00:36:34.010
That counts as a
subspace of itself.
00:36:34.010 --> 00:36:35.830
You always want to allow that.
00:36:35.830 --> 00:36:39.750
Then the others are lines --
00:36:39.750 --> 00:36:45.690
any line, meaning infinitely
far in both directions
00:36:45.690 --> 00:36:49.810
through the zero.
00:36:55.110 --> 00:36:57.790
So that's like
the whole space --
00:36:57.790 --> 00:37:00.550
that's like whole two D space.
00:37:00.550 --> 00:37:02.860
This is like one dimension.
00:37:02.860 --> 00:37:05.320
Is this line the same as R^1 ?
00:37:05.320 --> 00:37:07.470
No.
00:37:07.470 --> 00:37:11.200
You could say it
looks a lot like R^1.
00:37:11.200 --> 00:37:14.380
R^1 was just a line
and this is a line.
00:37:14.380 --> 00:37:17.460
But this is a line inside R^2.
00:37:17.460 --> 00:37:20.440
The vectors here
have two components.
00:37:20.440 --> 00:37:23.600
So that's not the same as R^1,
because there the vectors only
00:37:23.600 --> 00:37:25.570
have one component.
00:37:25.570 --> 00:37:29.590
Very close, you could
say, but not the same.
00:37:29.590 --> 00:37:30.320
Okay.
00:37:30.320 --> 00:37:32.250
And now there's a
third possibility.
00:37:36.550 --> 00:37:40.940
There's a third
subspace that's --
00:37:40.940 --> 00:37:47.970
of R^2 that's not the whole
thing, and it's not a line.
00:37:47.970 --> 00:37:50.170
It's even less.
00:37:50.170 --> 00:37:52.840
It's just the zero vector alone.
00:37:52.840 --> 00:37:55.170
The zero vector alone, only.
00:38:01.250 --> 00:38:05.550
I'll often call this
subspace Z, just for zero.
00:38:05.550 --> 00:38:07.700
Here's a line, L.
00:38:07.700 --> 00:38:10.010
Here's a plane, all of R^2.
00:38:10.010 --> 00:38:14.680
So, do you see that
the zero vector's okay?
00:38:14.680 --> 00:38:16.970
You would just -- to
understand subspaces,
00:38:16.970 --> 00:38:20.820
we have to know the rules -- and
knowing the rules means that we
00:38:20.820 --> 00:38:25.040
have to see that yes, the
zero vector by itself,
00:38:25.040 --> 00:38:27.990
just this guy alone
satisfies the rules.
00:38:27.990 --> 00:38:28.690
Why's that?
00:38:28.690 --> 00:38:31.320
Oh, it's too dumb to tell you.
00:38:31.320 --> 00:38:36.430
If I took that and added it
to itself, I'm still there.
00:38:36.430 --> 00:38:40.320
If I took that and multiplied
by seventeen, I'm still there.
00:38:40.320 --> 00:38:44.070
So I've done the operations,
adding and multiplying
00:38:44.070 --> 00:38:47.010
by numbers, that are
required, and I didn't go
00:38:47.010 --> 00:38:50.300
outside this one point space.
00:38:53.570 --> 00:38:57.170
So that's always -- that's
the littlest subspace.
00:38:57.170 --> 00:39:00.930
And the largest subspace is the
whole thing and in-between come
00:39:00.930 --> 00:39:02.370
all --
00:39:02.370 --> 00:39:04.080
whatever's in between.
00:39:04.080 --> 00:39:04.580
Okay.
00:39:04.580 --> 00:39:07.610
So for example, what's
in between for R^3?
00:39:07.610 --> 00:39:12.100
So if I'm in ordinary three
dimensions, the subspace is R,
00:39:12.100 --> 00:39:18.250
all of R^3 at one extreme,
the zero vector at the bottom.
00:39:18.250 --> 00:39:23.430
And then a plane, a
plane through the origin.
00:39:23.430 --> 00:39:26.510
Or a line, a line
through the origin.
00:39:26.510 --> 00:39:32.970
So with R^3, the subspaces were
R^3, plane through the origin,
00:39:32.970 --> 00:39:37.560
line through the origin and
a zero vector by itself,
00:39:37.560 --> 00:39:43.030
zero zero zero, just
that single vector.
00:39:43.030 --> 00:39:44.360
Okay, you've got the idea.
00:39:47.470 --> 00:39:51.080
But, now comes --
00:39:51.080 --> 00:39:53.350
the reality is --
00:39:53.350 --> 00:39:57.530
what are these -- where
do these subspaces come --
00:39:57.530 --> 00:40:00.950
how do they come
out of matrices?
00:40:00.950 --> 00:40:06.080
And I want to take
this matrix --
00:40:06.080 --> 00:40:08.350
oh, let me take that matrix.
00:40:08.350 --> 00:40:17.430
So I want to create some
subspaces out of that matrix.
00:40:17.430 --> 00:40:26.980
Well, one subspace
is from the columns.
00:40:26.980 --> 00:40:29.760
Okay.
00:40:29.760 --> 00:40:34.050
So this is the
important subspace,
00:40:34.050 --> 00:40:38.190
the first important subspace
that comes from that matrix --
00:40:38.190 --> 00:40:40.750
I'm going to -- let
me call it A again.
00:40:40.750 --> 00:40:44.370
Back to -- okay.
00:40:44.370 --> 00:40:48.150
I'm looking at the columns of A.
00:40:48.150 --> 00:40:50.530
Those are vectors in R^3.
00:40:50.530 --> 00:40:52.380
So the columns are in R^3.
00:40:52.380 --> 00:40:58.100
The columns are in R^3.
00:41:02.280 --> 00:41:04.585
So I want those columns
to be in my subspace.
00:41:08.970 --> 00:41:11.960
Now I can't just put two
columns in my subspace
00:41:11.960 --> 00:41:14.512
and call it a subspace.
00:41:14.512 --> 00:41:16.970
What do I have to throw in --
if I'm going to put those two
00:41:16.970 --> 00:41:21.460
columns in, what else has got
to be there to have a subspace?
00:41:21.460 --> 00:41:25.050
I must be able to
add those things.
00:41:25.050 --> 00:41:28.460
So the sum of those columns --
00:41:28.460 --> 00:41:34.970
so these columns are in R^3,
and I have to be able --
00:41:34.970 --> 00:41:37.330
I'm, you know, I want
that to be in my subspace,
00:41:37.330 --> 00:41:39.080
I want that to be
in my subspace,
00:41:39.080 --> 00:41:42.880
but therefore I have to be able
to multiply them by anything.
00:41:42.880 --> 00:41:45.910
Zero zero zero has got
to be in my subspace.
00:41:45.910 --> 00:41:48.630
I have to be able to add
them so that four five five
00:41:48.630 --> 00:41:50.150
is in the subspace.
00:41:50.150 --> 00:41:53.054
I've got to be able to add one
of these plus three of these.
00:41:53.054 --> 00:41:54.470
That'll give me
some other vector.
00:41:57.100 --> 00:42:02.180
I have to be able to take
all the linear combinations.
00:42:02.180 --> 00:42:14.200
So these are columns in R^3 and
all there linear combinations
00:42:14.200 --> 00:42:16.920
form a subspace.
00:42:21.260 --> 00:42:23.400
What do I mean by
linear combinations?
00:42:23.400 --> 00:42:26.060
I mean multiply
that by something,
00:42:26.060 --> 00:42:28.290
multiply that by
something and add.
00:42:28.290 --> 00:42:33.350
The two operations of linear
algebra, multiplying by numbers
00:42:33.350 --> 00:42:36.060
and adding vectors.
00:42:36.060 --> 00:42:38.930
And, if I include
all the results,
00:42:38.930 --> 00:42:40.875
then I'm guaranteed
to have a subspace.
00:42:43.570 --> 00:42:46.860
I've done the job.
00:42:46.860 --> 00:42:49.210
And we'll give it a name --
00:42:49.210 --> 00:42:49.960
the column space.
00:42:53.740 --> 00:42:54.380
Column space.
00:43:01.220 --> 00:43:05.920
And maybe I'll call it C of A.
00:43:05.920 --> 00:43:07.120
C for column space.
00:43:11.580 --> 00:43:15.020
There's an idea there that --
00:43:15.020 --> 00:43:22.750
Like, the central idea
for today's lecture is --
00:43:22.750 --> 00:43:25.220
got a few vectors.
00:43:25.220 --> 00:43:27.130
Not satisfied with
a few vectors,
00:43:27.130 --> 00:43:29.800
we want a space of vectors.
00:43:29.800 --> 00:43:33.160
The vectors, they're in --
these vectors in -- are in R^3 ,
00:43:33.160 --> 00:43:37.050
so our space of vectors
will be vectors in R^3.
00:43:37.050 --> 00:43:40.940
The key idea's -- we
have to be able to take
00:43:40.940 --> 00:43:42.400
their combinations.
00:43:42.400 --> 00:43:47.300
So tell me, geometrically,
if I drew all these things --
00:43:47.300 --> 00:43:50.060
like if I drew one two four,
that would be somewhere maybe
00:43:50.060 --> 00:43:50.930
there.
00:43:50.930 --> 00:43:54.740
If I drew three three one,
who knows, might be --
00:43:54.740 --> 00:43:57.140
I don't know, I'll say there.
00:43:57.140 --> 00:44:01.690
There's column one,
there's column two.
00:44:01.690 --> 00:44:06.700
What else -- what's in
the whole column space?
00:44:06.700 --> 00:44:11.160
How do I draw the
whole column space now?
00:44:11.160 --> 00:44:13.430
I take all combinations
of those two vectors.
00:44:15.970 --> 00:44:18.220
Do I get -- well, I
guess I actually listed
00:44:18.220 --> 00:44:19.160
the possibilities.
00:44:19.160 --> 00:44:21.940
Do I get the whole space?
00:44:21.940 --> 00:44:24.190
Do I get a plane?
00:44:24.190 --> 00:44:26.984
I get more than a
line, that's for sure.
00:44:26.984 --> 00:44:28.900
And I certainly get more
than the zero vector,
00:44:28.900 --> 00:44:31.610
but I do get the
zero vector included.
00:44:31.610 --> 00:44:34.160
What do I get if I combine --
00:44:34.160 --> 00:44:39.115
take all the combinations
of two vectors in R^3 ?
00:44:44.040 --> 00:44:46.450
So I've got all this stuff on --
00:44:46.450 --> 00:44:49.040
that whole line gets filled
out, that whole line gets filled
00:44:49.040 --> 00:44:51.190
out, but all in-between
gets filled out --
00:44:51.190 --> 00:44:52.900
between the two
lines because I --
00:44:52.900 --> 00:44:56.610
I allowed to add something
from one line, something
00:44:56.610 --> 00:44:57.850
from the other.
00:44:57.850 --> 00:44:58.810
You see what's coming?
00:44:58.810 --> 00:44:59.643
I'm getting a plane.
00:45:05.060 --> 00:45:06.790
That's my -- and it's
through the origin.
00:45:10.210 --> 00:45:17.950
Those two vectors, namely one
two four and three three one,
00:45:17.950 --> 00:45:20.590
when I take all
their combinations,
00:45:20.590 --> 00:45:21.770
I fill out a whole plane.
00:45:21.770 --> 00:45:25.240
Please think about that.
00:45:25.240 --> 00:45:28.280
That's the picture
you have to see.
00:45:28.280 --> 00:45:31.940
You sure have to see it in R^3
, because we're going to do it
00:45:31.940 --> 00:45:36.880
in R^10, and we may take a
combination of five vectors
00:45:36.880 --> 00:45:40.740
in R^10, and what will we have?
00:45:40.740 --> 00:45:41.630
God knows.
00:45:41.630 --> 00:45:44.910
It's some subspace.
00:45:44.910 --> 00:45:46.880
We'll have five vectors.
00:45:46.880 --> 00:45:49.010
They'll all have ten components.
00:45:49.010 --> 00:45:52.320
We take their combinations.
00:45:52.320 --> 00:45:58.240
We don't have R^5 , because our
vectors have ten components.
00:45:58.240 --> 00:46:05.020
And we possibly have, like,
some five dimensional flat thing
00:46:05.020 --> 00:46:06.680
going through the
origin for sure.
00:46:09.220 --> 00:46:12.110
Well, of course, if those five
vectors were all on the line,
00:46:12.110 --> 00:46:13.710
then we would only
get that line.
00:46:13.710 --> 00:46:16.840
So, you see, there are, like,
other possibilities here.
00:46:16.840 --> 00:46:21.690
It depends what -- it depends
on those five vectors.
00:46:21.690 --> 00:46:25.440
Just like if our two columns
had been on the same line,
00:46:25.440 --> 00:46:28.640
then the column space would
have been only a line.
00:46:28.640 --> 00:46:31.440
Here it was a plane.
00:46:31.440 --> 00:46:31.940
Okay.
00:46:35.700 --> 00:46:37.610
I'm going to stop at that point.
00:46:37.610 --> 00:46:44.220
That's the central idea of
-- the great example of how
00:46:44.220 --> 00:46:48.960
to create a subspace
from a matrix.
00:46:48.960 --> 00:46:51.990
Take its columns, take
their combinations,
00:46:51.990 --> 00:46:57.360
all their linear combinations
and you get the column space.
00:46:57.360 --> 00:47:01.060
And that's the
central sort of --
00:47:01.060 --> 00:47:04.320
we're looking at linear
algebra at a higher level.
00:47:04.320 --> 00:47:07.600
When I look at A -- now,
I want to look at Ax=b.
00:47:07.600 --> 00:47:10.090
That'll be the first
thing in the next lecture.
00:47:13.650 --> 00:47:17.300
How do I understand
Ax=b in this language --
00:47:17.300 --> 00:47:22.580
in this new language of vector
spaces and column spaces.
00:47:22.580 --> 00:47:24.830
And what are other subspaces?
00:47:24.830 --> 00:47:30.230
So the column space is a big
one, there are others to come.
00:47:30.230 --> 00:47:32.270
Okay, thanks.