OK.
This is linear algebra
lecture eleven.
And at the end of lecture ten,
I was talking about some vector
spaces, but they're --
the things in
those vector spaces
were not what we
usually call vectors.
Nevertheless, you
could add them and you
could multiply by numbers,
so we can call them vectors.
I think the example
I was working
with they were matrices.
So the -- so we had
like a matrix space,
the space of all three
by three matrices.
And I'd like to just pick
up on that, because --
we've been so specific about
n dimensional space here,
and you really want to see that
the same ideas work as long
as you can add and
multiply by scalars.
So these new, new vector
spaces, the example I took
was the space M of all
three by three matrices.
OK.
I can add them, I can
multiply by scalars.
I can multiply two of them
together, but I don't do that.
That's not part of the
vector space picture.
The vector space part is
just adding the matrices
and multiplying by numbers.
And that's fine, we stay
within this space of three
by three matrices.
And I had some subspaces
that were interesting,
like the symmetric, the
subspace of symmetric matrices,
symmetric three by threes.
Or the subspace of upper
triangular three by threes.
Now I, I use the word subspace
because it follows the rule.
If I add two symmetric
matrices, I'm still symmetric.
If I multiply two
symmetric matrices,
is the product
automatically symmetric?
No.
But I'm not
multiplying matrices.
I'm just adding.
So I'm fine.
This is a subspace.
Similarly, if I add two
upper triangular matrices,
I'm still upper triangular.
And, that's a subspace.
Now I just want to take these
as example and ask, well,
what's a basis
for that subspace?
What's the dimension
of that subspace?
And what's bd- dimension
of the whole space?
So, there's a natural basis for
all three by three matrices,
and why don't we
just write it down.
So, so M, a basis for M.
Again, all three by threes.
OK.
And then I'll just count how
many members are in that basis
and I'll know the dimension.
And OK, it's going to
take me a little time.
In fact, what is the dimension?
Any idea of what I'm
coming up with next?
How many numbers does it
take to specify that three
by three matrix?
Nine.
Nine is the, is the
dimension I'm going to find.
And the most obvious basis
would be the matrix that's
that matrix and then this
matrix with a one there
and that's two of them,
shall I put in the third one,
and then onwards, and
the last one maybe
would end with the one.
OK.
That's like the standard basis.
In fact, our space is
practically the same
as nine dimensional space.
It's just the nine numbers
are written in a square
instead of in a column.
But somehow it's different
and, and ought to be thought
of as --
natural for itself.
Because now what about the
symmetric three by threes?
So that's a subspace.
Just let's just think, what's
the dimension of that subspace
and what's a basis
for that subspace.
OK.
And I guess this
question occurs to me.
If I look at this subspace
of symmetric three
by threes, well, how many
of these original basis
members belong to the subspace?
I think only three of them do.
This one is symmetric.
This last one is symmetric.
And the one in the middle with
a, with a one in that position
-- in the two two position,
would be symmetric.
But so I've got three of these
original nine are symmetric,
but, so this is an
example where --
but that's, that's
not all, right?
What's the dimension?
Let's put the dimensions down.
Dimension of the,
of M, was nine.
What's the dimension of --
shall we call this S -- is what?
What's the dimension of this?
I'm sort of taking simple
examples where we can, we can,
spot the answer to
these questions.
So how many -- if I
have a symmetric --
think of all symmetric
matrices as a subspace,
how many parameters do I choose
in three by three symmetric
matrices?
Six, right.
If I choose the
diagonal that's three,
and the three entries
above the diagonal,
then I know what the
three entries below.
So the dimension is six.
I guess what's the
dimension of this here?
Let's call this space
U for upper triangular.
So what's the dimension of that
space of all upper triangular
three by threes?
Again six.
Again six.
And, but we haven't got a -- we
haven't seen -- well, actually,
maybe we have got a basis here
for, the upper triangulars.
I guess six of these guys,
one, two, three, four,
and a, and a couple more,
would be upper triangular.
So there's a accidental case
where the big basis contains
in it a basis for the subspace.
But with the symmetric
guy, it didn't have.
The symmetric guy the,
basis -- so you see --
a basis is the basis
for the big space,
we generally need to think it
all over again to get a basis
for the subspace.
And then how do I
get other subspaces?
Well, we spoke before about, the
subspace the symmetric matrices
and the upper triangular.
This is symmetric
and upper triangular.
What's the, what's the
dimension of that space?
OK.
Well, what's in that space?
So what's -- if a matrix
is symmetric and also upper
triangular, that
makes it diagonal.
So this is the same as
the diagonal matrices,
diagonal three by threes.
And the dimension of this,
of S intersect U, right --
you're OK with that symbol?
That's, that's the vectors
that are in both S and U,
and that's D.
So S intersect U
is the diagonals.
And the dimension of the
diagonal matrices is three.
And we've got a
basis, no problem.
OK, as I write that, I think,
OK, what about putting --
so this is like,
this intersection --
is taking all the
vectors that are
in both, that are symmetric
and also upper triangular.
Now we looked at the union.
Suppose I take the
matrices that are symmetric
or upper triangular.
What -- why was that no good?
So why is it no -- why I
not interested in the union,
putting together
those two subspaces?
So this, these are matrices that
are in S or in U, or possibly
both, so they, the
diagonals included.
But what's bad about this?
It's not a subspace.
It's like having,
taking, you know,
a couple of lines in the
plane and stopping there.
A line -- this is -- so there's
a three dimensional subspace
of a nine dimensional space,
there's -- ooh, sorry, six.
There's a six
dimensional subspace
of a nine dimensional space.
There's another one.
But they, they're headed
in different directions,
so we, we can't just
put them together.
We have to fill in.
So that's what we do.
To get this bigger space that
I'll write with a plus sign,
this is combinations of
things in S and things in U.
OK.
So that's the final space
I'm going to introduce.
I have a couple of subspaces.
I can take their intersection.
And now I'm interested in not
their union but their sum.
So this would be the,
this is the intersection,
and this will be their sum.
So what do I need
for a subspace here?
I take anything in S
plus anything in U.
I don't just take things
that are in S and pop
in also, separately,
things that are in U.
This is the sum of
any element of S,
that is, any symmetric
matrix, plus any
in U, any element of U.
OK.
Now as long as we've
got an example here,
tell me what we get.
If I take every
symmetric matrix,
take all symmetric
matrices, and add them
to all upper
triangular matrices,
then I've got a whole lot of
matrices and it is a subspace.
And what's -- it's
a vector space,
and what vector space
would I then have?
Any idea what,
what matrices can I
get out of a symmetric
plus an upper triangular?
I can get anything.
I get all matrices.
I get all three by threes.
It's worth thinking about that.
It's just like stretch your
mind a little, just a little,
to, to think of these subspaces
and what their intersection is
and what their sum is.
And now can I give you
a little -- oh, well,
let's figure out the dimension.
So what's the
dimension of S plus U?
In this example is nine, because
we got all three by threes.
So the original spaces had, the
original symmetric space had
dimension six and the original
upper triangular space
had dimension six.
And actually I'm seeing
here a nice formula.
That the dimension of S
plus the dimension of U --
if I have two subspaces,
the dimension of one plus
the dimension of the other
-- equals the dimension
of their intersection plus
the dimension of their sum.
Six plus six is three plus nine.
That's kind of satisfying, that
these natural operations --
and we've -- this
is it, actually,
this is the set of
natural things to do with,
with subspaces.
That, the dimensions
come out in a good way.
OK.
Maybe I'll take just one more
example of a vector space
that doesn't have vectors in it.
It's come from
differential equations.
So this is a one
more new vector space
that we'll give just
a few minutes to.
Suppose I have a differential
equation like d^2y/dx^2+ y=0.
OK.
I look at the solutions
to that equation.
So what are the solutions
to that equation?
y=cos(x) is a solution.
y=sin(x) is a solution.
y equals -- well, e to the (ix)
is a solution, if you want,
if you allow me to put that in.
But why should I put that in?
It's already there.
You see, I'm really looking
at a null space here.
I'm looking at the null space
of a differential equation.
That's the solution space.
And describe the solution
space, all solutions
to this differential equation.
So the equation is y''+y=0.
Cosine's, cosine's a
solution, sine is a solution.
Now tell me all the solutions.
They're -- so I
don't need e^(ix).
Forget that.
What are all the
complete solutions?
Is what?
A combination of these.
The complete
solution is y equals
some multiple of the cosine
plus some multiple of the sine.
That's a vector space.
That's a vector space.
What's the dimension
of that space?
What's a basis for that space?
OK, let me ask
you a basis first.
If I take the set of
solutions to that second order
differential equation --
there it is, those
are the solutions.
What's a basis for that space?
Now remember, what's
the, what question I
asking?
Because if you know the
question I'm asking,
you'll see the answer.
A basis means all
the guys in the space
are combinations of
these basis vectors.
Well, this is a basis.
sin x, cos x there is a basis.
Those two -- they're like
the special solutions, right?
We had special
solutions to Ax=b.
Now we've got special solutions
to differential equations.
Sorry, we had special
solutions to Ax=0, I misspoke.
The special solutions
were for the null space
just as here we're talking
about the null space.
Do you see that here
is a -- those two --
and what's the dimension
of the solution space?
How many vectors in this basis?
Two, the sine and cosine.
Are those the only
basis for this space?
By no means.
e^(ix) and e^(-ix)
would be another basis.
Lots of bases.
But do you see that really what
a course in differential --
in linear differential equations
is about is finding a basis
for the solution space.
The dimension of the solution
space will always be --
will be two, because we have
a second order equation.
So that's, like
there's 18.03 in --
five minutes of 18.06 is enough
to, to take care of 18.03.
So there's a -- that's
one more example.
OK.
And of course the
point of the example
is these things don't
look like vectors.
They look like functions.
But we can call them vectors,
because we can add them
and we can multiply
by constants,
so we can take
linear combinations.
That's all we have
to be allowed to do.
So that's really why this idea
of linear algebra and basis
and dimension and so on
plays a wider role than --
our constant discussions
of m by n matrices.
OK.
That's what I wanted to
say about that topic.
Now of course the key, number
associated with matrices,
to go back to that
number, is the rank.
And the rank, what do
we know about the rank?
Well, we know it's
not bigger than m
and it's not bigger than n.
So but I'd like to have a
little discussion on the rank.
Maybe I'll put that here.
So I'm picking up this
topic of rank one matrices.
And the reason I'm interested
in rank one matrices
is that they ought to be simple.
If the rank is only one, the
matrix can't get away from
us.
So for example, let me take
-- let me create a rank one
matrix.
OK.
Suppose it's three --
suppose it's two by three.
And let me give
you the first row.
What can the second row be?
Tell me a possible second row
here, for, for this matrix
to have rank one.
A possible second row is?
Two eight ten.
The second row is a
multiple of the first row.
It's not independent.
So tell me a basis
for the -- oh yeah,
sorry to keep bringing
up these same questions.
After the quiz I'll
stop, but for now,
tell me a basis
for the row space.
A basis for the row space of
that matrix is the first row,
right?
The first row, one four five.
A basis for the column
space of this matrix is?
What's the dimension
of the column space?
The dimension of the
column space is also one,
right?
Because it's also the rank.
The dimension -- you remember
the dimension of the column
space equals the rank equals the
dimension of the column space
of the transpose, which
is the row space of A.
OK, and in this case
it's one, r is one.
And sure enough, all
the columns are --
all the other columns are
multiples of that column.
Now there's -- there ought
to be a nice way to see that,
and here it is.
I can write that matrix as
its pivot column, one two,
times its --
times one four five.
A column times a row,
one column times one row
gives me a matrix, right?
If I multiply a
column by a row, that,
g- that's a two by one matrix
times a one by three matrix,
and the result of the
multiplication is two by three.
And it comes out right.
So what I want to -- my point
is the rank one matrices that
every rank one matrix has the
form some column times some
row.
So U is a column vector,
V is a column vector --
but I make it into a row
by putting in V transpose.
So that's the -- complete
picture of rank one matrices.
We'll be interested
in rank one matrices.
Later we'll find, oh, their
determinant, that'll be easy,
their eigenvalues,
that'll be interesting.
Rank one matrices are
like the building blocks
for all matrices.
And actually maybe
you can guess.
If I took any matrix, a five by
seventeen matrix of rank four,
then it seems pretty
likely -- and it's true,
that I could break that five
by seventeen matrix down
as a combination of
rank one matrices.
And probably how many
of those would I need?
If I have a five by seventeen
matrix of rank four,
I'll need four of them, right.
Four rank one matrices.
So the rank one matrices are
the, are the building blocks.
And out -- I can produce every,
I can produce every five by --
every rank four matrix out
of four rank one matrices.
That brings me to a
question, of course.
OK.
Would the rank four
matrices form a subspace?
Let me take all five by
seventeen matrices and think
about rank four -- the
subset of rank four matrices.
Let me -- I'll write this down.
You seem I'm reviewing
for the quiz,
because I'm asking the kind of
questions that are short enough
but -- that bring out do you
know what these words mean.
So I take --
my matrix space M now is all
five by seventeen matrices.
And now the question I ask is
the subset of, of rank four
matrices, is that a subspace?
If I add a matrix of -- so if I
multiply a matrix of rank four
by --
of rank four or less,
let's say, because I
have to let the zero matrix in
if it's going to be a subspace.
But, but that doesn't just
because the zero matrix
got in there doesn't
mean I have a subspace.
So if I -- so the, the question
really comes down to --
if I add two rank four
matrices, is the sum rank four?
What do you think?
If -- no, not usually.
Not usually.
If I add two rank four
matrices, the sum is probably --
what could I say about the sum?
Well, actually, well,
the rank could be five.
It's a general fact, actually,
that the rank of A plus B
can't be more than rank
of A plus the rank of B.
So this would say if
I added two of those,
the rank couldn't be larger
than eight, but I know actually
the rank couldn't be as
large as eight anyway.
What -- how big
could the rank be,
for, for the rank
of a matrix in M?
Could be as large as
five, right, right.
So they're all sort
of natural ideas.
So it's rank four matrices
or rank one matrices --
let me, let me change
that to rank one.
Let me take the subset
of rank one matrices.
Is that a vector space?
If I add a rank one matrix
to a rank one matrix?
No.
It's most likely going
to have rank two.
So this is --
So I'll just make that point.
Not a subspace.
OK.
OK.
Those are topics that
I wanted to, just
fill out the, the
previous lectures.
The I'll ask one more
subspace question, a,
a more, a more, likely example.
Suppose I'm in -- let me put,
put this example on a new
board.
Suppose I'm in R, in R^4.
So my typical vector in R^4 has
four components, v1, v2, v3,
and v4.
Suppose I take the
subspace of vectors
whose components add to zero.
So I let S be all v, all vectors
v in four dimensional space
with v1+v2+v3+v4=0.
So I just want to consider
that bunch of vectors.
Is it a subspace, first of all?
It is a subspace.
It is a subspace.
What's -- how do we see that?
It is a subspace.
I -- formally I should check.
If I have one vector that with
whose components add to zero
and I multiply that
vector by six --
the components still add to
zero, just six times as --
six times zero.
If I have a couple of v
and a w and I add them,
the, the components
still add to zero.
OK, it's a subspace.
What's the dimension
of that space
and what's a basis
for that space?
So you see how I can just
describe a space and we --
we can ask for the dimension
-- ask for the basis first
and the dimension.
Of course, the
dimension's the one
that's easy to tell
me in a single word.
What's the dimension
of our subspace S here?
And a basis tell me --
some vectors in it.
Well, I'm going to make ask you
again to guess the dimension.
Again I think I heard it.
The dimension is three.
Three.
Now how does this
connect to our Ax=0?
Is this the null
space of something?
Is that the null
space of a matrix?
And then we can
look at the matrix
and, and we know everything
about those subspaces.
This is the null
space of what matrix?
What's the matrix where the
null space is then Ab=0.
So I want this
equation to be Ab=0.
b is now the vector.
And what's the matrix that,
that we're seeing there?
It's the matrix of four ones.
Do you see that that's -- that
if I look at Ab=0 for this
matrix A, I multiply by b
and I get this requirement,
that the components add to zero.
So I'm really when
I speak about S --
I'm speaking about the
null space of that matrix.
OK.
Let's just say we've
got a matrix now,
we want its null space.
Well, we -- tell
me its rank first.
The rank of that
matrix is one, thanks.
So r is one.
What's the general
formula for the dimension
of the null space?
The dimension of the null
space of a matrix is --
in general, an m by
n matrix of rank r?
How many independent
guys in the null space?
n-r, right?
n-r.
In this case, n is
four, four columns.
The rank is one, so the null
space is three dimensions.
So of course y- you could
see it in this case,
but you can also see it
here in our systematic way
of dealing with the four
fundamental subspaces
of a matrix.
So what actually what,
what are all four subspaces
then?
The row space is clear.
The row space is in R^4.
Yeah, can we take the
four fundamental subspaces
of this matrix?
Let's just kill this example.
The row space is
one dimensional.
It's all multiples
of that, of that row.
The null space is
three dimensional.
Oh, you better give me a
basis for the null space.
So what's a basis
for the null space?
The special solutions.
To find the special solutions,
I look for the free variables.
The free variables here
are -- there's the pivot.
The free variables are
two, three, and four.
So the basis, basis
for S, for S will be --
I'm expecting three vectors,
three special solutions.
I give the value one
to that free variable,
and what's the pivot
variable if the --
this is going to
be a vector in S?
Minus one.
Now they're always added to
-- the entries add to zero.
The second special
solution has a one
in the second free variable,
and again a minus one
makes it right.
The third one has a one in
the third free variable,
and again a minus
one makes it right.
That's my answer.
That's the answer I
would be looking for.
The -- a basis for
this subspace S,
you would just
list three vectors,
and those would be the
natural three to list.
Not the only possible three,
but those are the special three.
OK, tell me about
the column space,
What's the column
space of this matrix A?
So the column space
is a subspace of R^1,
because m is only one.
The columns only
have one component.
So the column space of S, the
column space of A is somewhere
in the space R^1,
because we only have --
these columns are short.
And what is the
column space actually?
I just, it's just talking with
these words is what I'm doing.
The column space for
that matrix is R^1.
The column space
for that matrix is
all multiples of that column.
And all multiples
give you all of R^1.
And what's the, the
remaining fourth space,
the null space of A
transpose is what?
So we transpose A.
We look for combinations
of the columns
now that give zero
for A transpose.
And there aren't any.
The only thing, the only
combination of these rows
to give the zero row is
the zero combination.
OK.
So let's just check dimensions.
The null space has
dimension three.
The row space has dimension one.
Three plus one is four.
The column space
has dimension one,
and what's the
dimension of this, like,
smallest possible space?
What's the dimension
of the zero space?
It's a subspace.
Zero.
What else could it be?
I mean, let's -- we have to
take a reasonable answer --
and the only reasonable
answer is zero.
So one plus zero gives -- this
was n, the number of columns,
and this is m, the
number of rows.
And let's just, let
me just say again
then the, the, the subspace
that has only that one
point, that point is zero
dimensional, of course.
And the basis is empty, because
if the dimension is zero,
there shouldn't be
anybody in the basis.
So the basis of that smallest
subspace is the empty set.
And the number of members
in the empty set is zero,
so that's the dimension.
OK.
Good.
Now I have just five
minutes to tell you about --
well, actually, about some,
some, some, this is now,
this last topic of small
world graphs, and leads into,
a lecture about graphs
and linear algebra.
But let me tell you --
in these last minutes the
graph that I interested in.
It's the graph where
-- so what is a graph?
Better tell you that first.
OK.
What's a graph?
OK.
This isn't calculus.
We're not, I'm not thinking
of, like, some sine curve.
The word graph is used in
a completely different way.
It's a set of, a bunch
of nodes and edges,
edges connecting the nodes.
So I have nodes like
five nodes and edges --
I'll put in some edges, I
could put, include them all.
There's -- well, let me
put in a couple more.
There's a graph with five
nodes and one two three four
five six edges.
And some five by six
matrix is going to tell us
everything about that graph.
Let me leave that
matrix to next time
and tell you about the
question I'm interested in.
Suppose, suppose the
graph isn't just,
just doesn't have just five
nodes, but suppose every,
suppose every person
in this room is a node.
And suppose there's an
edge between two nodes
if those two people are friends.
So have I described a graph?
It's a pretty big graph,
hundred, hundred nodes.
And I don't know how
many edges are in there.
There's an edge
if you're friends.
So that's the graph
for this class.
A, a similar graph you could
take for the whole country,
so two hundred and
sixty million nodes.
And edges between friends.
And the question for that graph
is how many steps does it take
to get from anybody to anybody?
What two people are furthest
apart in this friendship graph,
say for the US?
By furthest apart, I
mean the distance from --
well, I'll tell you my
distance to Clinton.
It's two.
I happened to go to college
with somebody who knows Clinton.
I don't know him.
So my distance to Clinton is not
one, because I don't, happily
or not, don't know him.
But I know somebody who does.
He's a Senator and so
I presume he knows him.
OK.
I don't know what your --
well, what's your distance
to Clinton?
Well, not more
than three, right.
Actually, true.
You know me.
I take credit for reducing your
Clinton distance to three --
what's your distance to Monica.
Not, anybody below -- below
four is in trouble here.
Or maybe three, but, right.
So -- and what's Hillary's
distance to Monica?
I don't think we'd better
put that on tape here.
That's one or two, I guess.
Is that right?
I don't -- well, we won't,
think more about that.
So actually, the,
the real question
is what are large distances?
How, how far apart could
people be separated?
And roughly this number
six degrees of separation
has kind of appeared as the
movie title, as the book title,
and it's with this meaning.
That roughly speaking --
six might be a fairly --
not too many people.
If you sit next to
somebody on an airplane,
you get talking to them.
You begin to discuss mutual
friends to sort of find out,
OK, what connections
do you have,
and very often
you'll find you're
connected in, like, two
or three or four steps.
And you remark,
it's a small world,
and that's how this expression
small world came up.
But six, I don't know if you
could find -- if it took six,
I don't know if you would
successfully discover those six
in a, in an airplane
conversation.
But here's the math
question, and I'll
leave it for next,
for lecture twelve,
and do a lot of linear
algebra in lecture twelve.
But the interesting point is
that with a few shortcuts,
the distances come
down dramatically.
That, I mean, all your distances
to Clinton immediately drop
to three by taking
linear algebra.
That's, like, an extra bonus
for taking linear algebra.
And to understand mathematically
what it is about these graphs
--
or like the graphs of
the World Wide Web.
There's a fantastic graph.
So many people would like to
understand and model the web.
What the -- where the edges are
links and the nodes are, sites,
websites.
I'll leave you with that
graph, and I'll see you --
have a good weekend,
and see you on Monday.
