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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 is two point seven that
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talks about permutations,
which finished the previous
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lecture, and transposes,
which also came in the previous
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lecture.
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There's a little more to do
with those guys,
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permutations and transposes.
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But then the heart of the
lecture will be the beginning of
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what you could say is the
beginning of linear algebra,
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the beginning of real linear
algebra which is seeing a bigger
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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 is good,
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because we're coming to the
place 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 -- so these
permutations,
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those are matrices P and they
execute row exchanges.
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And we may need them.
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We may have a perfectly good
matrix, a perfect matrix A
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that's invertible that we can
solve A x=b, but to do it --
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I've got to allow myself that
extra freedom that if a zero
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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 want to collect the
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ideas together.
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And principle,
I could even have to do that
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two times, or more times.
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So I have to allow -- to
complete the -- the theory,
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the possibility that I take my
matrix A, I start elimination,
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I find out that I need row
exchanges and I do it and
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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 -- what happens to A
equal L U?
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So A equal L U -- this was a
matrix L with ones on the
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diagonal and zeroes above and
multipliers below,
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and this U we know,
with zeroes down here.
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That's only possible.
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That description of elimination
assumes that we don't have a P,
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that we don't have any row
exchanges.
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And now I just want to say,
okay, how do I account for row
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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, because it doesn't like
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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,
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which 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 will say,
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well, what do row exchanges do,
but we won't do them unless we
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have to.
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But we may have to.
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And then, the result is -- 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 -- it
gets the rows into the right
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order, 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, that a permutation
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matrix -- 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 -- and
then just leave this
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factorization 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 with
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reordered rows.
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I include in reordering the
possibility that you just leave
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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 -- well,
it gets a little -- it gets
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more interesting algebraically
if you've got four rows,
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you might exchange them all in
one big cycle.
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One to two, two to three,
three to four,
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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)...
(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 -- is just the same as
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the transpose.
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You might take a permutation
matrix, multiply by its
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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 -- we'll be highly
interested in matrices that have
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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 is the inverse of
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that.
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Okay.
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We'll be interested in matrices
that have P transpose P equal
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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 -- in the
center of these special
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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 by
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four permutations,
there are five factorial which
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is a hundred and twenty,
five times twenty four would
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bump us up to a hundred and
twenty --
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so listing all the five by five
permutations would be not so
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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
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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
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matrix 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 -- I'll use a T,
also Matlab would use a prime.
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And the result will be -- I'll
right it here,
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because this was three rows and
two columns, this was a three by
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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
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a row --
that column becomes the other
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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 -- 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 and I ask
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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 which
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was in row J,
column I.
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So the row and column -- the
row and column numbers just get
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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 -- 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, so whatever is
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sitting on the diagonal -- but
now what's above the diagonal,
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like a one, had better be
there, a seven had better be
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here, 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 --
I'm just asking you to notice
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this family of matrices that 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.
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This is the case where the
transpose gives the same matrix
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back again.
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That's totally simple to see.
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Okay.
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Could actually -- maybe I could
even say when would we get such
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a matrix?
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For example,
this -- that matrix is
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absolutely far from symmetric,
right?
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The transpose isn't even the
same shape --
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because it's rectangular,
it turns the -- lies down on
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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 call it R for
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rectangular?
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So let that be R for
rectangular matrix and let that
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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 and then ask you why,
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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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And I'm going to do it for that
particular R transpose R which
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was -- let's see,
the column was one two four
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three three one.
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I called that one R transpose,
didn't I, and I called this guy
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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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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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Next to that I'm getting four
and three, a seven.
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Now what do I get there?
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This eleven came from one three
times two three,
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right?
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Row one, column two.
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What goes here?
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Row two, column one.
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But no difference.
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One three two three or two
three one three,
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same thing.
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It's going to be an eleven.
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That's the symmetry.
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I can continue to fill it out.
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What -- oh, let's get that
seven.
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That seven will show up down
here, too, and then four more
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numbers.
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That seven will show up here
because one three times four one
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gave the seven,
but also four one times one
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three will give that seven.
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Do you see that it works?
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Actually, do you want to see it
work also in matrix language?
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I mean, that's quite
convincing, right?
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That seven is no accident.
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The eleven is no accident.
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But just tell me how do I know
if I transpose this guy --
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How do I know it's symmetric?
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Well, I'm going to transpose
it.
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And when I transpose it,
I'm hoping I get the matrix
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back again.
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So can I transpose R transpose
R?
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So just -- so,
why?
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Well, my suggestion is take the
transpose.
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That's the only way to show
it's symmetric.
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Take the transpose and see that
it didn't change.
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Okay, so I take the transpose
of R transpose R.
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Okay.
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How do I do that?
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This is our little practice on
the rules for transposes.
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So the rule for transposes is
the order gets reversed.
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Just like inverses,
which we did prove,
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same rule for transposes and --
which we'll now use.
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So the order gets reversed.
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It's the transpose of that that
comes first, and the transpose
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of this that comes -- no.
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Is that -- yeah.
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That's what I have to write,
right?
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This is a product of two
matrices and I want its
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00:19:04 --> 00:19:04
transpose.
283
00:19:04 --> 00:19:09
So I put the matrices in the
opposite order and I transpose
284
00:19:09 --> 00:19:09.77
them.
285
00:19:09.77 --> 00:19:11
But what have I got here?
286
00:19:11 --> 00:19:14
What is R transpose transpose?
287
00:19:14 --> 00:19:17
Well, don't all speak at once.
288
00:19:17 --> 00:19:23
R transpose transpose,
I flipped over the diagonal,
289
00:19:23 --> 00:19:28
I flipped over the diagonal
again, so I've got R.
290
00:19:28 --> 00:19:34
And that's just my point,
that if I started with this
291
00:19:34 --> 00:19:39.46
matrix, I transposed it,
I got it back again.
292
00:19:39.46 --> 00:19:44
So that's the check,
without using numbers,
293
00:19:44 --> 00:19:49
but with --
it checked in two lines that I
294
00:19:49 --> 00:19:53
always get symmetric matrices
this way.
295
00:19:53 --> 00:19:58
And actually,
that's where they come from in
296
00:19:58 --> 00:20:01.91
so many practical applications.
297
00:20:01.91 --> 00:20:02
Okay.
298
00:20:02 --> 00:20:08
So now I've said something
today about permutations and
299
00:20:08 --> 00:20:13
about transposes and about
symmetry and I'm ready for
300
00:20:13 --> 00:20:16
chapter three.
301
00:20:16 --> 00:20:24
Can we take a breath -- the
tape won't take a breath,
302
00:20:24 --> 00:20:32
but the lecturer will,
because to tell you about
303
00:20:32 --> 00:20:41
vector spaces is -- we really
have to start now and think,
304
00:20:41 --> 00:20:43
okay, listen up.
305
00:20:43 --> 00:20:47
What are vector spaces?
306
00:20:47 --> 00:20:50
And what are sub-spaces?
307
00:20:50 --> 00:20:51
Okay.
308
00:20:51 --> 00:20:59
So, the point is,
The main operations that we do
309
00:20:59 --> 00:21:02
-- what do we do with vectors?
310
00:21:02 --> 00:21:03
We add them.
311
00:21:03 --> 00:21:06
We know how to add two vectors.
312
00:21:06 --> 00:21:11
We multiply them by numbers,
usually called scalers.
313
00:21:11 --> 00:21:16
If we have a vector,
we know what three V is.
314
00:21:16 --> 00:21:22
If we have a vector V and W,
we know what V plus W is.
315
00:21:22 --> 00:21:27
Those are the two operations
that we've got to be able to do.
316
00:21:27 --> 00:21:31
To legitimately talk about a
space of vectors,
317
00:21:31 --> 00:21:36
the requirement is that we
should be able to add the things
318
00:21:36 --> 00:21:41.57
and multiply by numbers and that
there should be some decent
319
00:21:41.57 --> 00:21:43.71
rules satisfied.
320
00:21:43.71 --> 00:21:44
Okay.
321
00:21:44 --> 00:21:47.68
So let me start with examples.
322
00:21:47.68 --> 00:21:51.99
So I'm talking now about vector
spaces.
323
00:21:51.99 --> 00:21:56
And I'm going to start with
examples.
324
00:21:56 --> 00:22:01
Let me say again what this word
space is meaning.
325
00:22:01 --> 00:22:07
When I say that word space,
that means to me that I've got
326
00:22:07 --> 00:22:13
a bunch of vectors,
a space of vectors.
327
00:22:13 --> 00:22:17.08
But not just any bunch of
vectors.
328
00:22:17.08 --> 00:22:23
It has to be a space of vectors
-- has to allow me to do the
329
00:22:23 --> 00:22:26
operations that vectors are for.
330
00:22:26 --> 00:22:32
I have to be able to add
vectors and multiply by numbers.
331
00:22:32 --> 00:22:37
I have to be able to take
linear combinations.
332
00:22:37 --> 00:22:42
Well, where did we meet linear
combinations?
333
00:22:42 --> 00:22:45
We met them back in,
say in R^2.
334
00:22:45 --> 00:22:48.15
So there's a vector space.
335
00:22:48.15 --> 00:22:50
What's that vector space?
336
00:22:50 --> 00:22:56
So R two is telling me I'm
talking about real numbers and
337
00:22:56 --> 00:23:00
I'm talking about two real
numbers.
338
00:23:00 --> 00:23:08
So this is all two dimensional
vectors -- real,
339
00:23:08 --> 00:23:14
such as -- well,
I'm not going to be able to
340
00:23:14 --> 00:23:17
list them all.
341
00:23:17 --> 00:23:23
But let me put a few down.
|3; 2|, |0;0|,
342
00:23:23 --> 00:23:24
|pi; e|.
343
00:23:24 --> 00:23:25.45
So on.
344
00:23:25.45 --> 00:23:29
And it's natural -- okay.
345
00:23:29 --> 00:23:36.23
Let's see, I guess I should do
algebra first.
346
00:23:36.23 --> 00:23:45
Algebra means what can I do to
these vectors?
347
00:23:45 --> 00:23:46
I can add them.
348
00:23:46 --> 00:23:48
I can add that to that.
349
00:23:48 --> 00:23:50
And how do I do it?
350
00:23:50 --> 00:23:52
A component at a time,
of course.
351
00:23:52 --> 00:23:57
Three two added to zero zero
gives me, three two.
352
00:23:57 --> 00:23:58
Sorry about that.
353
00:23:58 --> 00:24:02
Three two added to pi e gives
me three plus pi,
354
00:24:02 --> 00:24:03
two plus e.
355
00:24:03 --> 00:24:06
Oh, you know what it does.
356
00:24:06 --> 00:24:10
And you know the picture that
goes with it.
357
00:24:10 --> 00:24:12
There's the vector three two.
358
00:24:12 --> 00:24:15
And often, the picture has an
arrow.
359
00:24:15 --> 00:24:19
The vector zero zero,
which is a highly important
360
00:24:19 --> 00:24:23
vector -- it's got,
like, the most important here
361
00:24:23 --> 00:24:23
-- is there.
362
00:24:23 --> 00:24:28.34
And of course there's not much
of an arrow.
363
00:24:28.34 --> 00:24:34
Pi -- I'll have to remember --
pi is about three and a little
364
00:24:34 --> 00:24:38
more, e is about two and a
little more.
365
00:24:38 --> 00:24:40
So maybe there's pi e.
366
00:24:40 --> 00:24:43
I never drew pi e before.
367
00:24:43 --> 00:24:48
It's just natural to -- this is
the first component on the
368
00:24:48 --> 00:24:53
horizontal and this is the
second component,
369
00:24:53 --> 00:24:56
going up the vertical.
370
00:24:56 --> 00:24:57
Okay.
371
00:24:57 --> 00:25:02
And the whole plane is R two.
372
00:25:02 --> 00:25:07
So R two is,
we could say,
373
00:25:07 --> 00:25:08
the plane.
374
00:25:08 --> 00:25:11
The xy plane.
375
00:25:11 --> 00:25:16
That's what everybody thinks.
376
00:25:16 --> 00:25:27
But the point is it's a vector
space because all those vectors
377
00:25:27 --> 00:25:29
are in there.
378
00:25:29 --> 00:25:38
If I removed one of them --
Suppose I removed zero zero.
379
00:25:38 --> 00:25:42
Suppose I tried to take the --
considered the X Y plane with a
380
00:25:42 --> 00:25:45
puncture, with a point removed.
381
00:25:45 --> 00:25:46
Like the origin.
382
00:25:46 --> 00:25:49
That would be,
like, awful to take the origin
383
00:25:49 --> 00:25:50
away.
384
00:25:50 --> 00:25:51
Why is that?
385
00:25:51 --> 00:25:53
Why do I need the origin there?
386
00:25:53 --> 00:25:58
Because I have to be allowed --
if I had these other vectors,
387
00:25:58 --> 00:26:03
I have to be allowed to
multiply three two -- this was
388
00:26:03 --> 00:26:06
three two -- by anything,
by any scaler,
389
00:26:06 --> 00:26:07
including zero.
390
00:26:07 --> 00:26:11
I've got to be allowed to
multiply by zero and the
391
00:26:11 --> 00:26:14
result's got to be there.
392
00:26:14 --> 00:26:17
I can't do without that point.
393
00:26:17 --> 00:26:21
And I have to be able to add
three two to the opposite guy,
394
00:26:21 --> 00:26:23
minus three minus two.
395
00:26:23 --> 00:26:27
And if I add those I'm back to
the origin again.
396
00:26:27 --> 00:26:30
No way I can do without the
origin.
397
00:26:30 --> 00:26:34
Every vector space has got that
zero vector in it.
398
00:26:34 --> 00:26:43.04
Okay, that's an easy vector
space, because we have a natural
399
00:26:43.04 --> 00:26:44
picture of it.
400
00:26:44 --> 00:26:45
Okay.
401
00:26:45 --> 00:26:48
Similarly easy is R^3.
402
00:26:48 --> 00:26:55
This would be all -- let me go
up a little here.
403
00:26:55 --> 00:27:03
This would be -- R three would
be all three dimensional vectors
404
00:27:03 --> 00:27:09
--
or shall I say vectors with
405
00:27:09 --> 00:27:12
three real components.
406
00:27:12 --> 00:27:13
Okay.
407
00:27:13 --> 00:27:23
Let me just to be sure we're
together, let me take the vector
408
00:27:23 --> 00:27:25
three two zero.
409
00:27:25 --> 00:27:30
Is that a vector in R^2 or R^3?
410
00:27:30 --> 00:27:34
Definitely it's in R^3.
411
00:27:34 --> 00:27:39
It's got three components.
412
00:27:39 --> 00:27:45
One of them happens to be zero,
but that's a perfectly okay
413
00:27:45 --> 00:27:45
number.
414
00:27:45 --> 00:27:48
So that's a vector in R^3.
415
00:27:48 --> 00:27:53
We don't want to mix up the --
I mean, keep these vectors
416
00:27:53 --> 00:27:56.93
straight and keep R^n straight.
417
00:27:56.93 --> 00:27:58
So what's R^n?
418
00:27:58 --> 00:27:58
R^n.
419
00:27:58 --> 00:28:03
So this is our big example,
is all vectors with n
420
00:28:03 --> 00:28:05
components.
421
00:28:05 --> 00:28:09
And I'm making these darn
things column vectors.
422
00:28:09 --> 00:28:14
Can I try to follow that
convention, that they'll be
423
00:28:14 --> 00:28:18
column vectors,
and their components should be
424
00:28:18 --> 00:28:19
real numbers.
425
00:28:19 --> 00:28:24.05
Later we'll need complex
numbers and complex vectors,
426
00:28:24.05 --> 00:28:25
but much later.
427
00:28:25 --> 00:28:25
Okay.
428
00:28:25 --> 00:28:29
So that's a vector space.
429
00:28:29 --> 00:28:30.35
Now, let's see.
430
00:28:30.35 --> 00:28:34
What do I have to tell you
about vector spaces?
431
00:28:34 --> 00:28:38
I said the most important
thing, which is that we can add
432
00:28:38 --> 00:28:42
any two of these and we -- still
in R^2.
433
00:28:42 --> 00:28:46
We can multiply by any number
and we're still in R^2.
434
00:28:46 --> 00:28:51
We can take any combination and
we're still in R^2.
435
00:28:51 --> 00:28:53
And same goes for R^n.
436
00:28:53 --> 00:29:00
It's -- honesty requires me to
mention that these operations of
437
00:29:00 --> 00:29:05.21
adding and multiplying have to
obey a few rules.
438
00:29:05.21 --> 00:29:11
Like, we can't just arbitrarily
say, okay, the sum of three two
439
00:29:11 --> 00:29:14
and pi e is zero zero.
440
00:29:14 --> 00:29:15
It's not.
441
00:29:15 --> 00:29:20
The sum of three two and minus
three two is zero zero.
442
00:29:20 --> 00:29:24
So -- oh, I'm not going to --
the book, actually,
443
00:29:24 --> 00:29:30
lists the eight rules that the
addition and multiplication have
444
00:29:30 --> 00:29:32
to satisfy, but they do.
445
00:29:32 --> 00:29:37
They certainly satisfy it in
R^n and usually it's not those
446
00:29:37 --> 00:29:41
eight rules that are in doubt.
447
00:29:41 --> 00:29:47
What's -- the question is,
can we do those additions and
448
00:29:47 --> 00:29:50
do we stay in the space?
449
00:29:50 --> 00:29:54
Let me show you a case where
you can't.
450
00:29:54 --> 00:30:00
So suppose this is going to be
not a vector space.
451
00:30:00 --> 00:30:06
Suppose I take the xy plane --
so there's R^2.
452
00:30:06 --> 00:30:09
That is a vector space.
453
00:30:09 --> 00:30:13
Now suppose I just take part of
it.
454
00:30:13 --> 00:30:14
Just this.
455
00:30:14 --> 00:30:22
Just this one -- this is one
quarter of the vector space.
456
00:30:22 --> 00:30:28.99
All the vectors with positive
or at least not negative
457
00:30:28.99 --> 00:30:31
components.
458
00:30:31 --> 00:30:33
Can I add those safely?
459
00:30:33 --> 00:30:33
Yes.
460
00:30:33 --> 00:30:37
If I add a vector with,
like, two -- three two to
461
00:30:37 --> 00:30:42
another vector like five six,
I'm still up in this quarter,
462
00:30:42 --> 00:30:44
no problem with adding.
463
00:30:44 --> 00:30:49
But there's a heck of a problem
with multiplying by scalers,
464
00:30:49 --> 00:30:54
because there's a lot of
scalers that will take me out of
465
00:30:54 --> 00:30:59
this quarter plane,
like negative ones.
466
00:30:59 --> 00:31:03
If I took three two and I
multiplied by minus five,
467
00:31:03 --> 00:31:05
I'm way down here.
468
00:31:05 --> 00:31:11
So that's not a vector space,
because it's not -- closed is
469
00:31:11 --> 00:31:12.42
the right word.
470
00:31:12.42 --> 00:31:16
It's not closed under
multiplication by all real
471
00:31:16 --> 00:31:17
numbers.
472
00:31:17 --> 00:31:23
So a vector space has to be
closed under multiplication and
473
00:31:23 --> 00:31:26
addition of vectors.
474
00:31:26 --> 00:31:28
In other words,
linear combinations.
475
00:31:28 --> 00:31:33
It -- so, it means that if I
give you a few vectors -- yeah
476
00:31:33 --> 00:31:38
look, here's an important --
here -- now we're getting to
477
00:31:38 --> 00:31:41
some really important vector
spaces.
478
00:31:41 --> 00:31:45
Well, R^n -- like,
they are the most important.
479
00:31:45 --> 00:31:51
But we will be interested in
so- in vector spaces that are
480
00:31:51 --> 00:31:52
inside R^n.
481
00:31:52 --> 00:31:59
Vector spaces that follow the
rules, but they -- we don't need
482
00:31:59 --> 00:32:04
all of -- see,
there we started with R^2 here,
483
00:32:04 --> 00:32:08
and took part of it and messed
it up.
484
00:32:08 --> 00:32:12
What we got was not a vector
space.
485
00:32:12 --> 00:32:18
Now tell me a vector space that
is part of R^2 and is still
486
00:32:18 --> 00:32:22
safely --
we can multiply,
487
00:32:22 --> 00:32:28
we can add and we stay in this
smaller vector space.
488
00:32:28 --> 00:32:32
So it's going to be called a
subspace.
489
00:32:32 --> 00:32:37
So I'm going to change this bad
example to a good one.
490
00:32:37 --> 00:32:38
Okay.
491
00:32:38 --> 00:32:44
So I'm going to start again
with R^2, but I'm going to take
492
00:32:44 --> 00:32:50
an example --
it is a vector space,
493
00:32:50 --> 00:32:55.9
so it'll be a vector space
inside R^2.
494
00:32:55.9 --> 00:33:01
And we'll call that a subspace
of R^2.
495
00:33:01 --> 00:33:02.25
Okay.
496
00:33:02.25 --> 00:33:04
What can I do?
497
00:33:04 --> 00:33:08
It's got something in it.
498
00:33:08 --> 00:33:13
Suppose it's got this vector in
it.
499
00:33:13 --> 00:33:15
Okay.
500
00:33:15 --> 00:33:19
If that vector's in my little
subspace and it's a true
501
00:33:19 --> 00:33:23
subspace, then there's got to be
some more in it,
502
00:33:23 --> 00:33:23
right?
503
00:33:23 --> 00:33:28
I have to be able to multiply
that by two, and that double
504
00:33:28 --> 00:33:29
vector has to be included.
505
00:33:29 --> 00:33:33
Have to be able to multiply by
zero, that vector,
506
00:33:33 --> 00:33:36
or by half, or by three
quarters.
507
00:33:36 --> 00:33:38
All these vectors.
508
00:33:38 --> 00:33:41
Or by minus a half,
or by minus one.
509
00:33:41 --> 00:33:46
I have to be able to multiply
by any number.
510
00:33:46 --> 00:33:51
So that is going to say that I
have to have that whole line.
511
00:33:51 --> 00:33:53
Do you see that?
512
00:33:53 --> 00:34:00
Once I get a vector in there --
I've got the whole line of all
513
00:34:00 --> 00:34:03
multiples of that vector.
514
00:34:03 --> 00:34:09.07
I can't have a vector space
without extending to get those
515
00:34:09.07 --> 00:34:11
multiples in there.
516
00:34:11 --> 00:34:14
Now I still have to check
addition.
517
00:34:14 --> 00:34:18
But that comes out okay.
518
00:34:18 --> 00:34:23.67
This line is going to work,
because I could add something
519
00:34:23.67 --> 00:34:29
on the line to something else on
the line and I'm still on the
520
00:34:29 --> 00:34:30
line.
521
00:34:30 --> 00:34:31.51
So, example.
522
00:34:31.51 --> 00:34:37
So this is all examples of a
subspace -- our example is a
523
00:34:37 --> 00:34:41.26
line in R^2 actually -- not just
any line.
524
00:34:41.26 --> 00:34:45
If I took this line,
would that -- so all the
525
00:34:45 --> 00:34:49
vectors on that line.
526
00:34:49 --> 00:34:56
So that vector and that vector
and this vector and this vector
527
00:34:56 --> 00:35:02
-- in lighter type,
I'm drawing something that
528
00:35:02 --> 00:35:03
doesn't work.
529
00:35:03 --> 00:35:06
It's not a subspace.
530
00:35:06 --> 00:35:13
The line in R^2 -- to be a
subspace, the line in R^2 must
531
00:35:13 --> 00:35:17
go through the zero vector.
532
00:35:17 --> 00:35:21
Because -- why is this line no
good?
533
00:35:21 --> 00:35:23
Let me do a dashed line.
534
00:35:23 --> 00:35:28
Because if I multiplied that
vector on the dashed line by
535
00:35:28 --> 00:35:33
zero, then I'm down here,
I'm not on the dashed line.
536
00:35:33 --> 00:35:35
Z- zero's got to be.
537
00:35:35 --> 00:35:40
Every subspace has got to
contain zero --
538
00:35:40 --> 00:35:45
because I must be allowed to
multiply by zero and that will
539
00:35:45 --> 00:35:47
always give me the zero vector.
540
00:35:47 --> 00:35:48
Okay.
541
00:35:48 --> 00:35:52
Now, I was going to make --
create some subspaces.
542
00:35:52 --> 00:35:56
Oh, while I'm in R^2,
why don't we think of all the
543
00:35:56 --> 00:35:57
possibilities.
544
00:35:57 --> 00:36:00
R two, there can't be that
many.
545
00:36:00 --> 00:36:04
So what are the possible
subspaces of R^2?
546
00:36:04 --> 00:36:07
Let me list them.
547
00:36:07 --> 00:36:12
So I'm listing now the
subspaces of R^2.
548
00:36:12 --> 00:36:20
And one possibility that we
always allow is all of R two,
549
00:36:20 --> 00:36:24
the whole thing,
the whole space.
550
00:36:24 --> 00:36:29
That counts as a subspace of
itself.
551
00:36:29 --> 00:36:35
You always want to allow that.
552
00:36:35 --> 00:36:43.24
Then the others are lines --
any line, meaning infinitely far
553
00:36:43.24 --> 00:36:47
in both directions through the
zero.
554
00:36:47 --> 00:36:55
So that's like the whole space
-- that's like whole two D
555
00:36:55 --> 00:36:56
space.
556
00:36:56 --> 00:36:59
This is like one dimension.
557
00:36:59 --> 00:37:03
Is this line the same as R^1 ?
558
00:37:03 --> 00:37:05.74
No.
559
00:37:05.74 --> 00:37:08.79
You could say it looks a lot
like R^1.
560
00:37:08.79 --> 00:37:11
R^1 was just a line and this is
a line.
561
00:37:11 --> 00:37:14
But this is a line inside R^2.
562
00:37:14 --> 00:37:17
The vectors here have two
components.
563
00:37:17 --> 00:37:22
So that's not the same as R^1,
because there the vectors only
564
00:37:22 --> 00:37:24
have one component.
565
00:37:24 --> 00:37:29
Very close, you could say,
but not the same.
566
00:37:29 --> 00:37:30
Okay.
567
00:37:30 --> 00:37:34
And now there's a third
possibility.
568
00:37:34 --> 00:37:42
There's a third subspace that's
-- of R^2 that's not the whole
569
00:37:42 --> 00:37:45
thing, and it's not a line.
570
00:37:45 --> 00:37:47
It's even less.
571
00:37:47 --> 00:37:52
It's just the zero vector
alone.
572
00:37:52 --> 00:37:55
The zero vector alone,
only.
573
00:37:55 --> 00:38:00
I'll often call this subspace
Z, just for zero.
574
00:38:00 --> 00:38:01
Here's a line,
L.
575
00:38:01 --> 00:38:04
Here's a plane,
all of R^2.
576
00:38:04 --> 00:38:09
So, do you see that the zero
vector's okay?
577
00:38:09 --> 00:38:16
You would just -- to understand
subspaces, we have to know the
578
00:38:16 --> 00:38:20
rules --
and knowing the rules means
579
00:38:20 --> 00:38:24
that we have to see that yes,
the zero vector by itself,
580
00:38:24 --> 00:38:27
just this guy alone satisfies
the rules.
581
00:38:27 --> 00:38:28
Why's that?
582
00:38:28 --> 00:38:30
Oh, it's too dumb to tell you.
583
00:38:30 --> 00:38:34
If I took that and added it to
itself, I'm still there.
584
00:38:34 --> 00:38:37
If I took that and multiplied
by seventeen,
585
00:38:37 --> 00:38:39
I'm still there.
586
00:38:39 --> 00:38:44
So I've done the operations,
adding and multiplying by
587
00:38:44 --> 00:38:49
numbers, that are required,
and I didn't go outside this
588
00:38:49 --> 00:38:50
one point space.
589
00:38:50 --> 00:38:54.6
So that's always -- that's the
littlest subspace.
590
00:38:54.6 --> 00:39:00
And the largest subspace is the
whole thing and in-between come
591
00:39:00 --> 00:39:03
all -- whatever's in between.
592
00:39:03 --> 00:39:03
Okay.
593
00:39:03 --> 00:39:07
So for example,
what's in between for R^3?
594
00:39:07 --> 00:39:12
So if I'm in ordinary three
dimensions, the subspace is R,
595
00:39:12 --> 00:39:16
all of R^3 at one extreme,
the zero vector at the bottom.
596
00:39:16 --> 00:39:20
And then a plane,
a plane through the origin.
597
00:39:20 --> 00:39:24
Or a line, a line through the
origin.
598
00:39:24 --> 00:39:28
So with R^3,
the subspaces were R^3,
599
00:39:28 --> 00:39:34
plane through the origin,
line through the origin and a
600
00:39:34 --> 00:39:39
zero vector by itself,
zero zero zero,
601
00:39:39 --> 00:39:42
just that single vector.
602
00:39:42 --> 00:39:45
Okay, you've got the idea.
603
00:39:45 --> 00:39:54
But, now comes -- the reality
is -- what are these -- where do
604
00:39:54 --> 00:40:02
these subspaces come -- how do
they come out of matrices?
605
00:40:02 --> 00:40:11
And I want to take this matrix
-- oh, let me take that matrix.
606
00:40:11 --> 00:40:19
So I want to create some
subspaces out of that matrix.
607
00:40:19 --> 00:40:25
Well, one subspace is from the
columns.
608
00:40:25 --> 00:40:27
Okay.
609
00:40:27 --> 00:40:34
So this is the important
subspace, the first important
610
00:40:34 --> 00:40:41
subspace that comes from that
matrix -- I'm going to -- let me
611
00:40:41 --> 00:40:44
call it A again.
612
00:40:44 --> 00:40:46
Back to -- okay.
613
00:40:46 --> 00:40:50
I'm looking at the columns of
A.
614
00:40:50 --> 00:40:53
Those are vectors in R^3.
615
00:40:53 --> 00:40:56
So the columns are in R^3.
616
00:40:56 --> 00:41:00
The columns are in R^3.
617
00:41:00 --> 00:41:04
So I want those columns to be
in my subspace.
618
00:41:04 --> 00:41:09
Now I can't just put two
columns in my subspace and call
619
00:41:09 --> 00:41:10
it a subspace.
620
00:41:10 --> 00:41:15
What do I have to throw in --
if I'm going to put those two
621
00:41:15 --> 00:41:20
columns in, what else has got to
be there to have a subspace?
622
00:41:20 --> 00:41:24
I must be able to add those
things.
623
00:41:24 --> 00:41:28
So the sum of those columns --
so these columns are in R^3,
624
00:41:28 --> 00:41:33
and I have to be able -- I'm,
you know, I want that to be in
625
00:41:33 --> 00:41:36
my subspace, I want that to be
in my subspace,
626
00:41:36 --> 00:41:41
but therefore I have to be able
to multiply them by anything.
627
00:41:41 --> 00:41:45
Zero zero zero has got to be in
my subspace.
628
00:41:45 --> 00:41:50
I have to be able to add them
so that four five five is in the
629
00:41:50 --> 00:41:51
subspace.
630
00:41:51 --> 00:41:57.33
I've got to be able to add one
of these plus three of these.
631
00:41:57.33 --> 00:42:00
That'll give me some other
vector.
632
00:42:00 --> 00:42:05
I have to be able to take all
the linear combinations.
633
00:42:05 --> 00:42:10
So these are columns in R^3 and
all there linear combinations
634
00:42:10 --> 00:42:13.59
form a subspace.
635
00:42:13.59 --> 00:42:17
What do I mean by linear
combinations?
636
00:42:17 --> 00:42:21.97
I mean multiply that by
something, multiply that by
637
00:42:21.97 --> 00:42:23
something and add.
638
00:42:23 --> 00:42:29
The two operations of linear
algebra, multiplying by numbers
639
00:42:29 --> 00:42:31
and adding vectors.
640
00:42:31 --> 00:42:36
And, if I include all the
results, then I'm guaranteed to
641
00:42:36 --> 00:42:39
have a subspace.
642
00:42:39 --> 00:42:43
I've done the job.
643
00:42:43 --> 00:42:52
And we'll give it a name -- the
column space.
644
00:42:52 --> 00:42:55
Column space.
645
00:42:55 --> 00:43:01
And maybe I'll call it C of A.
646
00:43:01 --> 00:43:05
C for column space.
647
00:43:05 --> 00:43:16
There's an idea there that --
Like, the central idea for
648
00:43:16 --> 00:43:19
today's lecture is -- got a few
vectors.
649
00:43:19 --> 00:43:24
Not satisfied with a few
vectors, we want a space of
650
00:43:24 --> 00:43:25.67
vectors.
651
00:43:25.67 --> 00:43:29
The vectors,
they're in -- these vectors in
652
00:43:29 --> 00:43:34
-- are in R^3 ,
so our space of vectors will be
653
00:43:34 --> 00:43:35
vectors in R^3.
654
00:43:35 --> 00:43:40
The key idea's --
we have to be able to take
655
00:43:40 --> 00:43:41
their combinations.
656
00:43:41 --> 00:43:45
So tell me, geometrically,
if I drew all these things --
657
00:43:45 --> 00:43:49
like if I drew one two four,
that would be somewhere maybe
658
00:43:49 --> 00:43:49
there.
659
00:43:49 --> 00:43:53
If I drew three three one,
who knows, might be -- I don't
660
00:43:53 --> 00:43:56
know, I'll say there.
661
00:43:56 --> 00:43:59
There's column one,
there's column two.
662
00:43:59 --> 00:44:03
What else -- what's in the
whole column space?
663
00:44:03 --> 00:44:07
How do I draw the whole column
space now?
664
00:44:07 --> 00:44:10
I take all combinations of
those two vectors.
665
00:44:10 --> 00:44:15
Do I get -- well,
I guess I actually listed the
666
00:44:15 --> 00:44:17
possibilities.
667
00:44:17 --> 00:44:19
Do I get the whole space?
668
00:44:19 --> 00:44:20
Do I get a plane?
669
00:44:20 --> 00:44:24
I get more than a line,
that's for sure.
670
00:44:24 --> 00:44:28.24
And I certainly get more than
the zero vector,
671
00:44:28.24 --> 00:44:31
but I do get the zero vector
included.
672
00:44:31 --> 00:44:37
What do I get if I combine --
take all the combinations of
673
00:44:37 --> 00:44:38
two vectors in R^3 ?
674
00:44:38 --> 00:44:43
So I've got all this stuff on
-- that whole line gets filled
675
00:44:43 --> 00:44:48
out, that whole line gets filled
out, but all in-between gets
676
00:44:48 --> 00:44:53
filled out -- between the two
lines because I --
677
00:44:53 --> 00:44:58
I allowed to add something from
one line, something from the
678
00:44:58 --> 00:44:59
other.
679
00:44:59 --> 00:45:01
You see what's coming?
680
00:45:01 --> 00:45:03
I'm getting a plane.
681
00:45:03 --> 00:45:06.9
That's my -- and it's through
the origin.
682
00:45:06.9 --> 00:45:11
Those two vectors,
namely one two four and three
683
00:45:11 --> 00:45:15
three one, when I take all their
combinations,
684
00:45:15 --> 00:45:18.9
I fill out a whole plane.
685
00:45:18.9 --> 00:45:21
Please think about that.
686
00:45:21 --> 00:45:24
That's the picture you have to
see.
687
00:45:24 --> 00:45:30
You sure have to see it in R^3
, because we're going to do it
688
00:45:30 --> 00:45:35
in R^10, and we may take a
combination of five vectors in
689
00:45:35 --> 00:45:39
R^10, and what will we have?
690
00:45:39 --> 00:45:40.05
God knows.
691
00:45:40.05 --> 00:45:41
It's some subspace.
692
00:45:41 --> 00:45:43.72
We'll have five vectors.
693
00:45:43.72 --> 00:45:46
They'll all have ten
components.
694
00:45:46 --> 00:45:48.67
We take their combinations.
695
00:45:48.67 --> 00:45:52
We don't have R^5 ,
because our vectors have ten
696
00:45:52 --> 00:45:53
components.
697
00:45:53 --> 00:45:57
And we possibly have,
like, some five dimensional
698
00:45:57 --> 00:46:02
flat thing going through the
origin for sure.
699
00:46:02 --> 00:46:06
Well, of course,
if those five vectors were all
700
00:46:06 --> 00:46:11
on the line, then we would only
get that line.
701
00:46:11 --> 00:46:13
So, you see,
there are, like,
702
00:46:13 --> 00:46:15
other possibilities here.
703
00:46:15 --> 00:46:20.8
It depends what --
it depends on those five
704
00:46:20.8 --> 00:46:21.61
vectors.
705
00:46:21.61 --> 00:46:27
Just like if our two columns
had been on the same line,
706
00:46:27 --> 00:46:32.04
then the column space would
have been only a line.
707
00:46:32.04 --> 00:46:34
Here it was a plane.
708
00:46:34 --> 00:46:34
Okay.
709
00:46:34 --> 00:46:37.71
I'm going to stop at that
point.
710
00:46:37.71 --> 00:46:43
That's the central idea of --
the great example of how to
711
00:46:43 --> 00:46:47
create a subspace from a matrix.
712
00:46:47 --> 00:46:50
Take its columns,
take their combinations,
713
00:46:50 --> 00:46:55
all their linear combinations
and you get the column space.
714
00:46:55 --> 00:46:59
And that's the central sort of
-- we're looking at linear
715
00:46:59 --> 00:47:01
algebra at a higher level.
716
00:47:01 --> 00:47:04
When I look at A -- now,
I want to look at Ax=b.
717
00:47:04 --> 00:47:08
That'll be the first thing in
the next lecture.
718
00:47:08 --> 00:47:12
How do I understand Ax=b in
this language -- in this new
719
00:47:12 --> 00:47:16
language of vector spaces and
column spaces.
720
00:47:16 --> 00:47:18
And what are other subspaces?
721
00:47:18 --> 00:47:22
So the column space is a big
one, there are others to come.
722
00:47:22 --> 00:47:25
Okay, thanks.