WEBVTT
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This is lecture twelve.
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OK.
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We've reached twelve lectures.
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And this one is more than
the others about applications
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of linear algebra.
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And I'll confess.
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When I'm giving you examples
of the null space and the row
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space, I create a little matrix.
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You probably see that I
just invent that matrix
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as I'm going.
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And I feel a little
guilty about it,
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because the truth is that real
linear algebra uses matrices
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that come from somewhere.
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They're not just, like, randomly
invented by the instructor.
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They come from applications.
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They have a definite structure.
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And anybody who works with
them gets, uses that structure.
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I'll just report,
like, this weekend
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I was at an event with
chemistry professors.
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OK, those guys are row
reducing matrices, and what
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matrices are they working with?
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Well, their little matrices tell
them how much of each element
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goes into the --
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or each molecule, how
many molecules of each
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go into a reaction
and what comes out.
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And by row reduction they
get a clearer picture
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of a complicated reaction.
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And this weekend I'm going to --
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to a sort of birthday
party at Mathworks.
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So Mathworks is out
Route 9 in Natick.
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That's where Matlab is created.
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It's a very, very
successful, software,
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tremendously successful.
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And the conference will be about
how linear algebra is used.
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And so I feel
better today to talk
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about what I think is
the most important model
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in applied math.
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And the discrete
version is a graph.
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So can I draw a graph?
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Write down the matrix
that's associated with it,
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and that's a great
source of matrices.
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You'll see.
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So a graph is
just, so a graph --
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to repeat -- has
nodes and edges.
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OK.
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And I'm going to write down
the graph, a graph, so I'm just
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creating a small graph here.
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As I mentioned last time,
we would be very interested
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in the graph of all, websites.
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Or the graph of all telephones.
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I mean -- or the graph of
all people in the world.
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Here let me take just,
maybe nodes one two three --
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well, I better put in an --
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I'll put in that edge and maybe
an edge to, to a node four,
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and another edge to node four.
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How's that?
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So there's a graph
with four nodes.
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So n will be four in my --
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n equal four nodes.
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And the matrix will have
m equal the number --
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there'll be a row
for every edge,
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so I've got one two
three four five edges.
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So that will be
the number of rows.
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And I have to to write down
the matrix that I want to,
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I want to study, I need to
give a direction to every edge,
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so I know a plus and
a minus direction.
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So I'll just do
that with an arrow.
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Say from one to two, one
to three, two to three,
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one to four, three to four.
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That just tells me, if I have
current flowing on these edges
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then I know whether it's
-- to count it as positive
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or negative according as whether
it's with the arrow or against
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the arrow.
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But I just drew those
arrows arbitrarily.
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OK.
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Because I -- my example is going
to come -- the example I'll --
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the words that I will use
will be words like potential,
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potential difference, currents.
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In other words, I'm
thinking of an electrical
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network.
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But that's just one possibility.
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My applied math class
builds on this example.
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It could be a hydraulic
network, so we could be doing,
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flow of water, flow of oil.
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Other examples, this
could be a structure.
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Like the -- a design
for a bridge or a design
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for a Buckminster Fuller dome.
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Or many other
possibilities, so many.
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So l- but let's take
potentials and currents
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as, as a basic
example, and let me
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create the matrix that tells
you exactly what the graph tells
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you.
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That's --, that's it.
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So now I'll call it the
incidence matrix, incidence
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matrix.
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So let me write it
down, and you'll see,
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OK.
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what its properties are.
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So every row
corresponds to an edge.
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I have five rows
from five edges,
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and let me write down again
what this graph looks like.
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OK, the first edge, edge one,
goes from node one to two.
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So I'm going to put in a minus
one and a plus one in th- this
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corresponds to node
one two three and four,
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That's a basis for the null
space. the four columns.
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The five rows correspond -- the
first row corresponds to edge
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one.
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Edge one leaves node one and
goes into node two, and that --
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and it doesn't touch
three and four.
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Edge two, edge two goes -- oh,
I haven't numbered these edges.
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I just figured that was probably
edge one, but I didn't say so.
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Let me take that to be edge one.
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Let me take this to be edge two.
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Let me take this
to be edge three.
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This is edge four.
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Ho, I'm discovering
-- no, wait a minute.
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Did I number that twice?
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Here's edge four.
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And here's edge five.
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All right.
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OK?
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So, so edge one, as I said,
goes from node one to two.
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Edge two goes from two to
three, node two to three, so
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minus one and one in the
second and third columns.
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Edge three goes
from one to three.
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I'm, I'm tempted to stop for a
moment with those three edges.
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The null space is actually one
dimensional, and it's the line
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Edges one two three,
those form what would we,
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A basis for the
null space will be
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just that.1 what do you call
the, the little, the little,
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the subgraph formed by
edges one, two, and three?
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That's a loop.
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And the number of loops and
the position of the loops
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will be crucial.
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OK.
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Actually, here's a
interesting point about loops.
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If I look at those
rows, corresponding
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to edges one two three,
and these guys made a loop.
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You want to tell me --
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if I just looked at
that much of the matrix
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it would be natural
for me to ask,
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are those rows independent?
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Are the rows independent?
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And can you tell from looking
at that if they are or are not
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independent?
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Do you see a, a relation
between those three rows?
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Yes.
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If I add that row to
that row, I get this row.
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So, so that's like a hint
here that loops correspond
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to dependent, linearly
dependent column --
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linearly dependent give me
a basis for the null space.
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rows.
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OK, let me complete
the incidence matrix.
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Number four, edge four is going
from node one to node four.
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And the fifth edge is going
from node three to node four.
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OK.
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There's my matrix.
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It came from the five
edges and the four nodes.
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And if I had a big graph,
I'd have a big matrix.
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And what questions do
I ask about matrices?
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Can I ask --
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here's the review now.
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There's a matrix that comes
from somewhere. of all vectors
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through that one.
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If, if it was a big graph,
it would be a large matrix,
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but a lot of zeros, right?
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Because every row only
has two non-zeros.
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So the number of -- it's
a very sparse matrix.
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The number of non-zeros
is exactly two times
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five, it's two m.
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Every row only
has two non-zeros.
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And that's with a
lot of structure.
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And -- that was the point
I wanted to begin with,
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that graphs, that real
graphs from real --
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real matrices from genuine
problems have structure.
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OK.
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We can ask, and because of
the structure, we can answer,
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If it -- yeah, let me
ask you just always,
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the, the main questions
about matrices.
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So first question, what
about the null space?
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So what I asking if I ask you
for the null space of that
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matrix?
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I'm asking you if I'm looking
at the columns of the matrix,
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four columns, and I'm asking
you, So there's a basis for it,
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and here is the whole null
are those columns independent?
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If the columns are independent,
then what's in the null space?
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Only the zero vector, right?
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The null space contains --
tells us what combinations
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of the columns -- it tells us
how to combine columns to get
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zero.
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Can -- and is there anything in
the null space of this matrix
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other than just the zero vector?
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In other words, are
those four columns
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independent or dependent?
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What else is in the null space?
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OK.
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That's our question.
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Let me, I don't know
if you see the answer.
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Whether there's -- so let's see.
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I guess we could do it properly.
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space.
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We could solve Ax=0.
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So let me solve Ax=0
to find the null space.
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OK.
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What's Ax?
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Can I put x in here
in, in little letters?
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x1, x2, x3, x4, that's
-- it's got four columns.
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Ax now is that matrix times x.
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And what do I get for Ax?
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If the camera can keep that
matrix multiplication there,
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I'll put the answer here.
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Ax equal -- what's the
first component of Ax?
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Can you take that first row,
minus one one zero zero,
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and multiply by the x, and
of course you get x2-x1.
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space.
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The second row, I get x3-x2.
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From the third row, I get x3-x1.
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Any multiple of one one one
one, it's the whole line in four
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From the fourth
row, I get x4-x1.
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And from the fifth
row, I get x4-x3.
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And I want to know
when is the thing zero.
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This is my equation, Ax=0.
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Notice what that
matrix A is doing,
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what we've created a
matrix that computes
00:13:19.350 --> 00:13:22.200
the differences across
every edge, the differences
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in potential. differences are
zero, and that x is in the null
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Let me even begin to
give this interpretation.
00:13:30.750 --> 00:13:38.930
I'm going to think of this
vector x, which is x1 x2 x3 x4,
00:13:38.930 --> 00:13:42.840
as the potentials at the nodes.
00:13:42.840 --> 00:13:45.760
So I'm introducing a word,
potentials at the nodes.
00:13:50.450 --> 00:13:57.600
And now if I multiply
by A, I get these --
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I get these five components,
x2-x1, et cetera.
00:14:04.730 --> 00:14:07.210
And what are they?
00:14:07.210 --> 00:14:09.130
They're potential differences.
00:14:09.130 --> 00:14:11.350
That's what A computes.
00:14:11.350 --> 00:14:14.530
If I have potentials at the
nodes and I multiply by A,
00:14:14.530 --> 00:14:18.520
it gives me the potential
differences, the differences
00:14:18.520 --> 00:14:21.735
in potential, across the edges.
00:14:25.870 --> 00:14:26.810
OK.
00:14:26.810 --> 00:14:29.830
When are those
differences all zero?
00:14:29.830 --> 00:14:33.310
So I'm looking for
the null space.
00:14:33.310 --> 00:14:37.810
Of course, if all the (x)s
are zero then I get zero.
00:14:37.810 --> 00:14:40.900
That, that just tells me,
of course, the zero vector
00:14:40.900 --> 00:14:43.240
is in the null space.
00:14:43.240 --> 00:14:47.870
But w- there's more
in the null space.
00:14:47.870 --> 00:14:51.040
Those columns are -- of
A are dependent, right --
00:14:51.040 --> 00:14:54.790
because I can find
solutions to that equation.
00:14:54.790 --> 00:14:56.172
dimensional space.
00:14:56.172 --> 00:14:57.255
Tell me -- the null space.
00:15:00.620 --> 00:15:04.010
Tell me one vector in the
null space, so tell me an x,
00:15:04.010 --> 00:15:10.490
it's got four components,
and it makes that thing zero.
00:15:10.490 --> 00:15:13.890
So what's a good x to do that?
00:15:13.890 --> 00:15:22.000
One one one one,
constant potential.
00:15:22.000 --> 00:15:52.130
If the potentials are constant,
then all the potential
00:15:52.130 --> 00:16:03.430
Do you see that
that's the null space?
00:16:03.430 --> 00:16:12.260
So the, so the dimension of
the null space of A is one.
00:16:12.260 --> 00:16:14.690
And there's a basis
for it, and there's
00:16:14.690 --> 00:16:16.190
everything that's in it.
00:16:16.190 --> 00:16:17.000
Good.
00:16:17.000 --> 00:16:23.250
And what does that
mean physically?
00:16:23.250 --> 00:16:25.240
I mean, what does that
mean in the application?
00:16:25.240 --> 00:16:26.850
That guy in the null space.
00:16:26.850 --> 00:16:33.170
It means that the
potentials can only
00:16:33.170 --> 00:16:36.560
be determined up to a constant.
00:16:36.560 --> 00:16:39.560
Potential differences are
what make current flow.
00:16:39.560 --> 00:16:40.970
That's what makes things happen.
00:16:45.360 --> 00:16:47.140
It's these potential
differences that
00:16:47.140 --> 00:16:50.320
will make something move
in the, in our network,
00:16:50.320 --> 00:16:53.950
between x2- between
node two and node one.
00:16:53.950 --> 00:16:57.650
Nothing will move if all
potentials are the same.
00:16:57.650 --> 00:17:01.710
If all potentials are c, c, c,
and c, then nothing will move.
00:17:01.710 --> 00:17:06.190
So we're, we have
this one parameter,
00:17:06.190 --> 00:17:10.060
this arbitrary constant
that raises or drops
00:17:10.060 --> 00:17:11.359
all the potentials.
00:17:11.359 --> 00:17:14.339
It's like ranking
football teams, whatever.
00:17:14.339 --> 00:17:16.510
We have a, there's a,
there's a constant --
00:17:16.510 --> 00:17:20.540
or looking at
temperatures, you know,
00:17:20.540 --> 00:17:23.770
there's a flow of heat from
higher temperature to lower
00:17:23.770 --> 00:17:26.349
temperature.
00:17:26.349 --> 00:17:28.840
If temperatures are
equal there's no flow,
00:17:28.840 --> 00:17:30.700
and therefore we can measure --
00:17:30.700 --> 00:17:36.950
we can measure temperatures
by, Celsius or we
00:17:36.950 --> 00:17:40.050
can start at absolute zero.
00:17:40.050 --> 00:17:44.000
And that arbitrary -- it's the
same arbitrary constant that,
00:17:44.000 --> 00:17:46.660
that was there in calculus.
00:17:46.660 --> 00:17:49.030
In calculus, right,
when you took
00:17:49.030 --> 00:17:53.050
the integral, the indefinite
integral, there was a plus c,
00:17:53.050 --> 00:17:58.030
and you had to set a starting
point to know what that c was.
00:17:58.030 --> 00:18:02.370
So here what often happens is
we fix one of the potentials,
00:18:02.370 --> 00:18:06.370
like the last one.
00:18:06.370 --> 00:18:12.050
So a typical thing would
be to ground that node.
00:18:12.050 --> 00:18:16.040
To set its potential at zero.
00:18:16.040 --> 00:18:19.070
And if we do that, if
we fix that potential
00:18:19.070 --> 00:18:25.570
so it's not unknown anymore,
then that column disappears
00:18:25.570 --> 00:18:29.440
and we have three columns,
and those three columns
00:18:29.440 --> 00:18:30.420
are independent.
00:18:30.420 --> 00:18:33.630
So I'll leave the
column in there,
00:18:33.630 --> 00:18:35.950
but we'll remember
that grounding a node
00:18:35.950 --> 00:18:38.520
is the way to get it out.
00:18:38.520 --> 00:18:42.680
And grounding a node is the
way to -- setting a node --
00:18:42.680 --> 00:18:47.890
setting a potential to zero
tells us the, the base for all
00:18:47.890 --> 00:18:48.510
potentials.
00:18:48.510 --> 00:18:51.370
Then we can compute the others.
00:18:51.370 --> 00:18:55.070
But what's the -- now
I've talked enough to ask
00:18:55.070 --> 00:18:58.420
OK. what the rank
of the matrix is?
00:18:58.420 --> 00:19:01.240
What's the rank then?
00:19:01.240 --> 00:19:03.000
The rank of the matrix.
00:19:03.000 --> 00:19:06.030
So we have a five
by four matrix.
00:19:06.030 --> 00:19:11.680
We've located its null
space, one dimensional.
00:19:11.680 --> 00:19:13.870
How many independent
columns do we have?
00:19:13.870 --> 00:19:15.050
What's the rank?
00:19:15.050 --> 00:19:17.880
It's three.
00:19:17.880 --> 00:19:21.180
And the first three columns,
or actually any three columns,
00:19:21.180 --> 00:19:22.920
will be independent.
00:19:22.920 --> 00:19:28.800
Any three potentials are
independent, good variables.
00:19:28.800 --> 00:19:35.490
The fourth potential
is not, we need to set,
00:19:35.490 --> 00:19:38.160
and typically we
ground that node.
00:19:38.160 --> 00:19:38.660
OK.
00:19:38.660 --> 00:19:40.270
Rank is three.
00:19:40.270 --> 00:19:43.820
Rank equals three.
00:19:43.820 --> 00:19:45.200
OK.
00:19:45.200 --> 00:19:48.420
Let's see, do I want to ask
you about the column space?
00:19:48.420 --> 00:19:52.360
The column space is all
combinations of those columns.
00:19:52.360 --> 00:19:55.560
I could say more
about it and I will.
00:19:55.560 --> 00:20:01.870
Let me go to the null
space of A transpose,
00:20:01.870 --> 00:20:07.680
because the equation A
transpose y equals zero
00:20:07.680 --> 00:20:10.090
is probably the most
fundamental equation
00:20:10.090 --> 00:20:11.410
of applied mathematics.
00:20:11.410 --> 00:20:14.860
All right, let's
talk about that.
00:20:14.860 --> 00:20:17.230
That deserves our attention.
00:20:17.230 --> 00:20:21.590
A transpose y equals zero.
00:20:21.590 --> 00:20:29.510
Let's -- let me put it on here.
00:20:29.510 --> 00:20:32.991
So A transpose y equals zero.
00:20:32.991 --> 00:20:33.490
OK.
00:20:33.490 --> 00:20:38.360
So now I'm finding the
null space of A transpose.
00:20:38.360 --> 00:20:41.200
Oh, and if I ask
you its dimension,
00:20:41.200 --> 00:20:43.760
you could tell me what it is.
00:20:43.760 --> 00:20:49.140
What's the dimension of the
null space of A transpose?
00:20:49.140 --> 00:20:51.640
We now know enough to
answer that question.
00:20:51.640 --> 00:20:55.030
What's the general formula for
the dimension of the null space
00:20:55.030 --> 00:20:55.770
of A transpose?
00:20:58.760 --> 00:21:02.560
A transpose, let me even
write out A transpose.
00:21:02.560 --> 00:21:10.070
This A transpose will
be n by m, right?
00:21:10.070 --> 00:21:12.770
In this case, it'll
be four by five.
00:21:12.770 --> 00:21:13.960
n by m.
00:21:13.960 --> 00:21:16.320
Those columns will become rows.
00:21:16.320 --> 00:21:25.480
Minus one zero minus one minus
one zero is now the first row.
00:21:25.480 --> 00:21:31.710
The second row of the matrix,
one minus one and three zeros.
00:21:31.710 --> 00:21:36.210
The third column now becomes
the third row, zero one one
00:21:36.210 --> 00:21:38.480
zero minus one.
00:21:38.480 --> 00:21:41.905
And the fourth column
becomes the fourth row.
00:21:45.730 --> 00:21:46.560
OK, good.
00:21:46.560 --> 00:21:48.400
There's A transpose.
00:21:48.400 --> 00:21:55.965
That multiplies y,
y1 y2 y3 y4 and y5.
00:22:00.740 --> 00:22:01.410
OK.
00:22:01.410 --> 00:22:03.830
Now you've had time to
think about this question.
00:22:03.830 --> 00:22:09.370
What's the dimension of the
null space, if I set all those
00:22:09.370 --> 00:22:10.705
-- wow.
00:22:13.410 --> 00:22:15.960
Usually -- sometime
during this semester,
00:22:15.960 --> 00:22:19.570
I'll drop one of these
erasers behind there.
00:22:19.570 --> 00:22:20.880
That's a great moment.
00:22:20.880 --> 00:22:22.570
There's no recovery.
00:22:22.570 --> 00:22:29.390
There's -- centuries of
erasers are back there.
00:22:29.390 --> 00:22:29.890
OK.
00:22:35.100 --> 00:22:38.020
OK, what's the dimension
of the null space?
00:22:40.640 --> 00:22:42.510
Give me the general
formula first
00:22:42.510 --> 00:22:46.050
in terms of r and m and n.
00:22:46.050 --> 00:22:49.580
This is like crucial, you --
00:22:49.580 --> 00:22:54.200
we struggled to, to decide
what dimension meant,
00:22:54.200 --> 00:22:59.980
and then we figured out
what it equaled for an m
00:22:59.980 --> 00:23:05.850
by n matrix of rank r, and
the answer was m-r, right?
00:23:05.850 --> 00:23:14.200
There are m=5 components,
m=5 columns of A transpose.
00:23:14.200 --> 00:23:18.400
And r of those columns
are pivot columns,
00:23:18.400 --> 00:23:19.960
because it'll have r pivots.
00:23:19.960 --> 00:23:21.410
It has rank r.
00:23:21.410 --> 00:23:28.090
And m-r are the free
ones now for A transpose,
00:23:28.090 --> 00:23:32.060
so that's five minus
three, so that's two.
00:23:35.040 --> 00:23:39.400
And I would like to
find this null space.
00:23:39.400 --> 00:23:41.800
I know its dimension.
00:23:41.800 --> 00:23:45.440
Now I want to find
out a basis for it.
00:23:45.440 --> 00:23:48.780
And I want to understand
what this equation is.
00:23:48.780 --> 00:23:53.500
So let me say what A transpose
y actually represents, why I'm
00:23:53.500 --> 00:23:57.120
interested in that equation.
00:23:57.120 --> 00:24:04.090
I'll put it down with those
old erasers and continue this.
00:24:04.090 --> 00:24:07.430
Here's the great picture
of applied mathematics.
00:24:07.430 --> 00:24:09.560
So let me complete that.
00:24:09.560 --> 00:24:14.040
There's a matrix
that I'll call C
00:24:14.040 --> 00:24:17.830
that connects potential
differences to currents.
00:24:17.830 --> 00:24:21.840
So I'll call these -- these
are currents on the edges,
00:24:21.840 --> 00:24:27.990
y1 y2 y3 y4 and y5.
00:24:27.990 --> 00:24:30.340
Those are currents on the edges.
00:24:34.160 --> 00:24:39.090
And this relation between
current and potential
00:24:39.090 --> 00:24:41.820
difference is Ohm's Law.
00:24:41.820 --> 00:24:43.340
This here is Ohm's Law.
00:24:47.060 --> 00:24:50.300
Ohm's Law says that
the current on an edge
00:24:50.300 --> 00:24:56.080
is some number times
the potential drop.
00:24:56.080 --> 00:24:59.820
That's -- and that number is
the conductance of the edge,
00:24:59.820 --> 00:25:01.290
one over the resistance.
00:25:01.290 --> 00:25:08.850
This is the old current
is, is, the relation
00:25:08.850 --> 00:25:13.960
of current, resistance,
and change in potential.
00:25:13.960 --> 00:25:17.760
So it's a change in potential
that makes some current happen,
00:25:17.760 --> 00:25:22.170
and it's Ohm's Law that says
how much current happens.
00:25:22.170 --> 00:25:22.820
OK.
00:25:22.820 --> 00:25:25.940
And then the final
step of this framework
00:25:25.940 --> 00:25:31.070
is the equation A
transpose y equals zero.
00:25:33.950 --> 00:25:38.990
And that's -- what
is that saying?
00:25:38.990 --> 00:25:40.470
It has a famous name.
00:25:40.470 --> 00:25:50.590
It's Kirchoff's Current Law,
KCL, Kirchoff's Current Law,
00:25:50.590 --> 00:25:53.030
A transpose y equals zero.
00:25:53.030 --> 00:25:55.950
So that when I'm solving, and
when I go back up with this
00:25:55.950 --> 00:26:03.770
blackboard and solve A
transpose y equals zero,
00:26:03.770 --> 00:26:05.950
it's this pattern of --
00:26:05.950 --> 00:26:08.110
that I want you to see.
00:26:08.110 --> 00:26:11.360
That we had rectangular
matrices, but --
00:26:11.360 --> 00:26:17.070
and real applications, but in
those real applications comes A
00:26:17.070 --> 00:26:18.500
and A transpose.
00:26:18.500 --> 00:26:22.030
So our four subspaces are
exactly the right things
00:26:22.030 --> 00:26:23.461
to know about.
00:26:23.461 --> 00:26:23.960
All right.
00:26:23.960 --> 00:26:28.030
Let's know about that
null space of A transpose.
00:26:28.030 --> 00:26:31.990
Wait a minute, where'd it go?
00:26:31.990 --> 00:26:33.190
There it is.
00:26:33.190 --> 00:26:34.680
OK.
00:26:34.680 --> 00:26:35.720
OK.
00:26:35.720 --> 00:26:38.210
Null space of A transpose.
00:26:38.210 --> 00:26:40.070
We know what its
dimension should be.
00:26:43.340 --> 00:26:47.670
Let's find out -- tell
me a vector in it.
00:26:47.670 --> 00:26:50.150
Tell me -- now, so
what I asking you?
00:26:50.150 --> 00:26:53.850
I'm asking you for
five currents that
00:26:53.850 --> 00:26:57.320
satisfy Kirchoff's Current Law.
00:26:57.320 --> 00:26:59.650
So we better understand
what that law says.
00:26:59.650 --> 00:27:01.780
That, that law, A
transpose y equals
00:27:01.780 --> 00:27:09.430
zero, what does that say, say
in the first row of A transpose?
00:27:09.430 --> 00:27:13.100
That says -- the so the first
row of A transpose says minus
00:27:13.100 --> 00:27:19.280
y1 minus y3 minus y4 is zero.
00:27:22.740 --> 00:27:25.020
Where did that
equation come from?
00:27:25.020 --> 00:27:27.470
Let me -- I'll redraw the graph.
00:27:27.470 --> 00:27:31.600
Can I redraw the graph here,
so that we -- maybe here,
00:27:31.600 --> 00:27:34.900
so that we see again --
00:27:34.900 --> 00:27:39.470
there was node one,
node two, node three,
00:27:39.470 --> 00:27:41.750
node four was off here.
00:27:41.750 --> 00:27:45.310
That was, that was our graph.
00:27:45.310 --> 00:27:47.220
We had currents on those.
00:27:47.220 --> 00:27:50.650
We had a current y1 going there.
00:27:50.650 --> 00:27:53.120
We had a current y --
what were the other,
00:27:53.120 --> 00:27:58.900
what are those edge numbers?
y4 here and y3 here.
00:28:01.480 --> 00:28:04.990
And then a y2 and a y5.
00:28:04.990 --> 00:28:07.860
I'm, I'm just copying what
was on the other board
00:28:07.860 --> 00:28:10.470
so it's ea-
convenient to see it.
00:28:10.470 --> 00:28:15.260
What is this equation telling
me, this first equation
00:28:15.260 --> 00:28:19.230
of Kirchoff's Current Law?
00:28:19.230 --> 00:28:21.900
What does that mean
for that graph?
00:28:21.900 --> 00:28:29.320
Well, I see y1, y3, and y4 as
the currents leaving node one.
00:28:29.320 --> 00:28:32.930
So sure enough, the first
equation refers to node one,
00:28:32.930 --> 00:28:34.720
and what does it say?
00:28:34.720 --> 00:28:39.430
It says that the
net flow is zero.
00:28:39.430 --> 00:28:43.230
That, that equation A transpose
y, Kirchoff's Current Law,
00:28:43.230 --> 00:28:47.860
is a balance equation,
a conservation law.
00:28:47.860 --> 00:28:51.770
Physicists, be overjoyed,
right, by this stuff.
00:28:51.770 --> 00:28:56.520
It, it says that in equals out.
00:28:56.520 --> 00:29:01.770
And in this case, the three
arrows are all going out,
00:29:01.770 --> 00:29:05.070
so it says y1, y3,
and y4 add to zero.
00:29:05.070 --> 00:29:07.280
Let's take the next one.
00:29:07.280 --> 00:29:16.150
The second row is y1-y2, and
that's all that's in that row.
00:29:16.150 --> 00:29:19.790
And that must have something
to do with node two.
00:29:19.790 --> 00:29:25.780
And sure enough, it says y1=y2,
current in equals current out.
00:29:25.780 --> 00:29:33.300
The third one, y2 plus
y3 minus y5 equals
00:29:33.300 --> 00:29:34.200
zero.
00:29:34.200 --> 00:29:38.340
That certainly will be
what's up at the third node.
00:29:38.340 --> 00:29:43.350
y2 coming in, y3 coming in,
y5 going out has to balance.
00:29:43.350 --> 00:29:48.770
And finally, y4
plus y5 equals zero
00:29:48.770 --> 00:29:57.760
says that at this node, y4
plus y5, the total flow,
00:29:57.760 --> 00:30:01.530
We don't -- you know, charge
doesn't accumulate at is zero.
00:30:01.530 --> 00:30:03.000
the nodes.
00:30:03.000 --> 00:30:06.180
It travels around.
00:30:06.180 --> 00:30:07.030
OK.
00:30:07.030 --> 00:30:09.940
Now give me --
00:30:09.940 --> 00:30:12.580
I come back now to the
linear algebra question.
00:30:12.580 --> 00:30:17.310
What's a vector y that
solves these equations?
00:30:17.310 --> 00:30:19.860
Can I figure out
what the null space
00:30:19.860 --> 00:30:28.470
is for this matrix, A transpose,
by looking at the graph?
00:30:28.470 --> 00:30:33.070
I'm happy if I don't
have to do elimination.
00:30:33.070 --> 00:30:35.780
I can do elimination,
we know how to do,
00:30:35.780 --> 00:30:39.110
we know how to find
the null space basis.
00:30:39.110 --> 00:30:42.580
We can do elimination
on this matrix,
00:30:42.580 --> 00:30:48.150
and we'll get it into a good
reduced row echelon form,
00:30:48.150 --> 00:30:51.330
and the special solutions
will pop right out.
00:30:51.330 --> 00:30:56.040
But I would like to --
even to do it without that.
00:30:56.040 --> 00:30:59.820
Let me just ask you first,
if I did elimination
00:30:59.820 --> 00:31:06.780
on that, on that, matrix, what
would the last row become?
00:31:06.780 --> 00:31:10.220
What would the last row -- if I
do elimination on that matrix,
00:31:10.220 --> 00:31:16.380
the last row of R will
be all zeros, right?
00:31:16.380 --> 00:31:17.190
Why?
00:31:17.190 --> 00:31:19.770
Because the rank is three.
00:31:19.770 --> 00:31:22.720
We only going to
have three pivots.
00:31:22.720 --> 00:31:26.540
And the fourth row will be
all zeros when we eliminate.
00:31:26.540 --> 00:31:32.020
So elimination will tell us
what, what we spotted earlier,
00:31:32.020 --> 00:31:36.160
what's the null space -- all
the, all the information,
00:31:36.160 --> 00:31:38.450
what are the dependencies.
00:31:38.450 --> 00:31:42.790
We'll find those by elimination,
but here in a real example,
00:31:42.790 --> 00:31:44.830
we can find them by thinking.
00:31:44.830 --> 00:31:46.030
OK.
00:31:46.030 --> 00:31:52.320
Again, my question is,
what is a solution y?
00:31:52.320 --> 00:31:55.940
How could current travel
around this network
00:31:55.940 --> 00:32:02.440
without collecting any
charge at the nodes?
00:32:02.440 --> 00:32:03.650
Tell me a y.
00:32:03.650 --> 00:32:04.540
OK.
00:32:04.540 --> 00:32:12.650
So a basis for the null
space of A transpose.
00:32:12.650 --> 00:32:15.980
How many vectors I looking for?
00:32:15.980 --> 00:32:17.220
Two.
00:32:17.220 --> 00:32:18.750
It's a two dimensional space.
00:32:18.750 --> 00:32:21.350
My basis should have
two vectors in it.
00:32:21.350 --> 00:32:23.570
Give me one.
00:32:23.570 --> 00:32:24.750
One set of currents.
00:32:24.750 --> 00:32:28.470
Suppose, let me start it.
00:32:28.470 --> 00:32:31.801
Let me start with y1 as one.
00:32:31.801 --> 00:32:32.300
OK.
00:32:32.300 --> 00:32:38.860
So one unit of -- one amp
travels on edge one with
00:32:38.860 --> 00:32:40.120
the arrow.
00:32:40.120 --> 00:32:41.480
OK, then what?
00:32:41.480 --> 00:32:42.275
What is y2?
00:32:44.790 --> 00:32:47.000
It's one also, right?
00:32:47.000 --> 00:32:50.230
And of course what you did was
solve Kirchoff's Current Law
00:32:50.230 --> 00:32:52.770
quickly in the second equation.
00:32:52.770 --> 00:32:53.600
OK.
00:32:53.600 --> 00:32:57.400
Now we've got one amp leaving
node one, coming around to node
00:32:57.400 --> 00:32:57.900
three.
00:32:57.900 --> 00:32:58.990
What shall we do now?
00:33:01.920 --> 00:33:05.350
Well, what shall I take
for y3 in other words?
00:33:05.350 --> 00:33:08.870
Oh, I've got a choice, but
why not make it what you said,
00:33:08.870 --> 00:33:11.530
negative one.
00:33:11.530 --> 00:33:16.150
So I have just sent current,
one amp, around that loop.
00:33:19.130 --> 00:33:23.000
What shall y4 and
y5 be in this case?
00:33:23.000 --> 00:33:25.050
We could take them to be zero.
00:33:25.050 --> 00:33:31.410
This satisfies
Kirchoff's Current Law.
00:33:31.410 --> 00:33:36.190
We could check it patiently,
that minus y1 minus y3
00:33:36.190 --> 00:33:37.050
gives zero.
00:33:37.050 --> 00:33:38.920
We know y1 is y2.
00:33:38.920 --> 00:33:42.710
The others, y4 plus
y5 is certainly zero.
00:33:42.710 --> 00:33:46.880
Any current around
a loop satisfies --
00:33:46.880 --> 00:33:49.090
satisfies the Current Law.
00:33:49.090 --> 00:33:52.140
Now you know how
to get another one.
00:33:52.140 --> 00:33:53.150
OK.
00:33:53.150 --> 00:33:55.890
Take current around this loop.
00:33:55.890 --> 00:34:04.650
So now let y3 be one, y5 be
one, and y4 be minus one.
00:34:04.650 --> 00:34:10.400
And so, so we have the first
basis vector sent current
00:34:10.400 --> 00:34:13.260
around that loop, the
second basis vector
00:34:13.260 --> 00:34:14.500
sends current around that
00:34:14.500 --> 00:34:15.590
loop.
00:34:15.590 --> 00:34:18.219
And I've -- and those
are independent,
00:34:18.219 --> 00:34:23.980
and I've got two solutions --
two vectors in the null space
00:34:23.980 --> 00:34:28.650
of A transpose, two solutions
to Kirchoff's Current Law.
00:34:28.650 --> 00:34:31.590
Of course you would
say what about sending
00:34:31.590 --> 00:34:34.830
current around the big loop.
00:34:34.830 --> 00:34:36.889
What about that vector?
00:34:36.889 --> 00:34:44.560
One for y1, one for y2,
nothing f- on y3, one for y5,
00:34:44.560 --> 00:34:46.889
and minus one for y4.
00:34:46.889 --> 00:34:48.690
What about that?
00:34:48.690 --> 00:34:52.630
Is that, is that in the
null space of A transpose?
00:34:52.630 --> 00:34:53.860
Sure.
00:34:53.860 --> 00:35:01.790
So why don't we now have a
third vector in the basis?
00:35:01.790 --> 00:35:05.890
Because it's not
independent, right?
00:35:05.890 --> 00:35:07.430
It's not independent.
00:35:07.430 --> 00:35:10.910
This vector is the
sum of those two.
00:35:10.910 --> 00:35:14.200
If I send current around
that and around that --
00:35:14.200 --> 00:35:18.940
then on this edge y3 it's going
to cancel out and I'll have
00:35:18.940 --> 00:35:23.080
altogether current around
the whole, the outside loop.
00:35:23.080 --> 00:35:24.700
That's what this
one is, but it's
00:35:24.700 --> 00:35:28.240
a combination of those two.
00:35:28.240 --> 00:35:33.300
Do you see that I've now, I've
identified the null space of A
00:35:33.300 --> 00:35:36.260
transpose --
00:35:36.260 --> 00:35:40.640
but more than that, we've
solved Kirchoff's Current Law.
00:35:43.160 --> 00:35:48.590
And understood it in
terms of the network.
00:35:48.590 --> 00:35:49.180
OK.
00:35:49.180 --> 00:35:53.010
So that's the null
space of A transpose.
00:35:53.010 --> 00:35:58.030
I guess I -- there's always one
more space to ask you about.
00:35:58.030 --> 00:36:04.370
Let's see, I guess I need the
row space of A, the column
00:36:04.370 --> 00:36:05.440
space of A transpose.
00:36:10.550 --> 00:36:14.240
So what's N, what's
its dimension?
00:36:14.240 --> 00:36:15.800
Yup?
00:36:15.800 --> 00:36:18.450
What's the dimension
of the row space of A?
00:36:18.450 --> 00:36:21.672
If I look at the original
A, it had five rows.
00:36:21.672 --> 00:36:22.755
How many were independent?
00:36:27.220 --> 00:36:30.620
Oh, I guess I'm asking
you the rank again, right?
00:36:30.620 --> 00:36:33.490
And the answer is three, right?
00:36:33.490 --> 00:36:35.610
Three independent rows.
00:36:35.610 --> 00:36:38.760
When I transpose it, there's
three independent columns.
00:36:38.760 --> 00:36:42.650
Are those columns
independent, those three?
00:36:42.650 --> 00:36:45.620
The first three columns,
are they the pivot columns
00:36:45.620 --> 00:36:46.720
of the matrix?
00:36:46.720 --> 00:36:48.180
No.
00:36:48.180 --> 00:36:51.450
Those three columns
are not independent.
00:36:51.450 --> 00:36:56.000
There's a in fact, this tells
me a relation between them.
00:36:56.000 --> 00:36:57.940
There's a vector in
the null space that
00:36:57.940 --> 00:37:01.530
says the first column
plus the second column
00:37:01.530 --> 00:37:03.400
equals the third column.
00:37:03.400 --> 00:37:07.490
They're not independent
because they come from a loop.
00:37:07.490 --> 00:37:11.570
So the pivot columns, the
pivot columns of this matrix
00:37:11.570 --> 00:37:18.630
will be the first, the second,
not the third, but the fourth.
00:37:18.630 --> 00:37:24.360
One, columns one,
two, and four are OK.
00:37:24.360 --> 00:37:28.170
Where are they -- those are
the columns of A transpose,
00:37:28.170 --> 00:37:30.350
those correspond to edges.
00:37:30.350 --> 00:37:35.340
So there's edge one,
there's edge two,
00:37:35.340 --> 00:37:37.600
and there's edge four.
00:37:42.370 --> 00:37:46.930
So there's a -- that's like --
00:37:46.930 --> 00:37:49.040
is a, smaller graph.
00:37:49.040 --> 00:37:52.860
If I just look at the part
of the graph that I've, that
00:37:52.860 --> 00:37:56.610
I've, thick -- used
with thick edges,
00:37:56.610 --> 00:38:00.240
it has the same four nodes.
00:38:00.240 --> 00:38:03.400
It only has three edges.
00:38:03.400 --> 00:38:08.520
And the, those edges correspond
to the independent guys.
00:38:08.520 --> 00:38:14.970
And in the graph there --
those three edges have no loop,
00:38:14.970 --> 00:38:15.900
right?
00:38:15.900 --> 00:38:19.550
The independent ones are the
ones that don't have a loop.
00:38:19.550 --> 00:38:22.820
All the -- dependencies
came from loops.
00:38:22.820 --> 00:38:25.800
They were the things in the
null space of A transpose.
00:38:25.800 --> 00:38:28.140
If I take three
pivot columns, there
00:38:28.140 --> 00:38:31.620
are no dependencies among
them, and they form a graph
00:38:31.620 --> 00:38:34.630
without a loop, and I
just want to ask you
00:38:34.630 --> 00:38:37.640
what's the name for a
graph without a loop?
00:38:37.640 --> 00:38:43.590
So a graph without a loop is
-- has got not very many edges,
00:38:43.590 --> 00:38:44.520
right?
00:38:44.520 --> 00:38:48.180
I've got four nodes and
it only has three edges,
00:38:48.180 --> 00:38:53.360
and if I put another edge
in, I would have a loop.
00:38:53.360 --> 00:38:56.390
So it's this graph
with no loops,
00:38:56.390 --> 00:39:01.670
and it's the one where the
rows of A are independent.
00:39:01.670 --> 00:39:04.400
And what's a graph
called that has no loops?
00:39:04.400 --> 00:39:06.810
It's called a tree.
00:39:06.810 --> 00:39:11.980
So a tree is the name for
a graph with no loops.
00:39:17.290 --> 00:39:24.400
And just to take
one last step here.
00:39:24.400 --> 00:39:28.170
Using our formula for dimension.
00:39:28.170 --> 00:39:33.710
Using our formula for
dimension, let's look --
00:39:33.710 --> 00:39:41.370
once at this formula.
00:39:41.370 --> 00:39:49.560
The dimension of the null
space of A transpose is m-r.
00:39:49.560 --> 00:39:51.230
OK.
00:39:51.230 --> 00:39:58.720
This is the number of loops,
number of independent loops.
00:39:58.720 --> 00:40:00.300
m is the number of edges.
00:40:04.250 --> 00:40:05.210
And what is r?
00:40:08.780 --> 00:40:12.460
What is r for our -- we'll
have to remember way back.
00:40:12.460 --> 00:40:17.970
The rank came -- from looking
at the columns of our matrix.
00:40:17.970 --> 00:40:19.950
So what's the rank?
00:40:19.950 --> 00:40:21.130
Let's just remember.
00:40:21.130 --> 00:40:26.710
Rank was -- you remember
there was one --
00:40:26.710 --> 00:40:29.750
we had a one dimensional
-- rank was n minus one,
00:40:29.750 --> 00:40:33.690
that's what I'm
struggling to say.
00:40:33.690 --> 00:40:37.810
Because there were n columns
coming from the n nodes,
00:40:37.810 --> 00:40:45.030
so it's minus, the number
of nodes minus one,
00:40:45.030 --> 00:40:50.070
because of that C, that
one one one one vector
00:40:50.070 --> 00:40:51.920
in the null space.
00:40:51.920 --> 00:40:54.210
The columns were
not independent.
00:40:54.210 --> 00:40:58.190
There was one dependency,
so we needed n minus one.
00:40:58.190 --> 00:41:01.610
This is a great formula.
00:41:01.610 --> 00:41:06.230
This is like the
first shall I, --
00:41:06.230 --> 00:41:09.410
write it slightly differently?
00:41:09.410 --> 00:41:14.890
The number of edges --
let me put things --
00:41:14.890 --> 00:41:17.940
have I got it right?
00:41:17.940 --> 00:41:22.630
Number of edges is m, the
number -- r- is m-r, OK.
00:41:22.630 --> 00:41:24.500
So, so I'm getting --
00:41:24.500 --> 00:41:27.070
let me put the number of
nodes on the other side.
00:41:27.070 --> 00:41:31.130
So I -- the number of nodes --
00:41:31.130 --> 00:41:34.890
I'll move that to
the other side --
00:41:34.890 --> 00:41:46.320
minus the number of edges
plus the number of loops is --
00:41:46.320 --> 00:41:50.060
I have minus, minus one is one.
00:41:50.060 --> 00:41:52.190
The number of nodes
minus the number
00:41:52.190 --> 00:41:56.030
of edges plus the
number of loops is one.
00:41:56.030 --> 00:41:58.460
These are like zero
dimensional guys.
00:41:58.460 --> 00:42:01.230
They're the points on the graph.
00:42:01.230 --> 00:42:03.570
The edges are like one
dimensional things,
00:42:03.570 --> 00:42:06.480
they're, they connect nodes.
00:42:06.480 --> 00:42:09.420
The loops are like two
dimensional things.
00:42:09.420 --> 00:42:12.240
They have, like, an area.
00:42:12.240 --> 00:42:16.030
And this count works
for every graph.
00:42:16.030 --> 00:42:22.950
And it's known as
Euler's Formula.
00:42:22.950 --> 00:42:26.770
We see Euler again,
that guy never stopped.
00:42:26.770 --> 00:42:28.940
OK.
00:42:28.940 --> 00:42:33.400
And can we just check
-- so what I saying?
00:42:33.400 --> 00:42:37.340
I'm saying that linear algebra
proves Euler's Formula.
00:42:37.340 --> 00:42:43.650
Euler's Formula is this great
topology fact about any graph.
00:42:43.650 --> 00:42:45.660
I'll draw, let me
draw another graph,
00:42:45.660 --> 00:42:51.980
let me draw a graph with
more edges and loops.
00:42:51.980 --> 00:42:53.610
Let me put in lots of --
00:42:53.610 --> 00:42:54.530
OK.
00:42:54.530 --> 00:42:56.500
I just drew a graph there.
00:42:56.500 --> 00:42:58.990
So what are the, what are the
quantities in that formula?
00:42:58.990 --> 00:43:00.870
How many nodes have I got?
00:43:00.870 --> 00:43:01.630
Looks like five.
00:43:04.180 --> 00:43:05.730
How many edges have I got?
00:43:05.730 --> 00:43:10.700
One two three four
five six seven.
00:43:10.700 --> 00:43:12.150
How many loops have I got?
00:43:12.150 --> 00:43:15.010
One two three.
00:43:15.010 --> 00:43:19.810
And Euler's right,
I always get one.
00:43:19.810 --> 00:43:27.700
That, this formula, is extremely
useful in understanding
00:43:27.700 --> 00:43:31.560
the relation of these quantities
-- the number of nodes,
00:43:31.560 --> 00:43:34.780
the number of edges,
and the number of loops.
00:43:34.780 --> 00:43:35.970
OK.
00:43:35.970 --> 00:43:39.890
Just complete this
lecture by completing
00:43:39.890 --> 00:43:42.310
this picture, this cycle.
00:43:42.310 --> 00:43:45.320
So let me come to the --
00:43:50.610 --> 00:43:57.200
so this expresses the
equations of applied math.
00:43:57.200 --> 00:43:59.800
This, let me call these
potential differences,
00:43:59.800 --> 00:44:04.100
say, E. So E is A x.
00:44:04.100 --> 00:44:08.010
That's the equation
for this step.
00:44:08.010 --> 00:44:11.890
The currents come from
the potential differences.
00:44:11.890 --> 00:44:15.020
y is C E.
00:44:15.020 --> 00:44:20.230
The potential -- the currents
satisfy Kirchoff's Current Law.
00:44:20.230 --> 00:44:23.230
Those are the equations of --
00:44:23.230 --> 00:44:25.290
with no source terms.
00:44:25.290 --> 00:44:32.590
Those are the equations of
electrical circuits of many --
00:44:32.590 --> 00:44:37.940
those are like the, the
most basic three equations.
00:44:37.940 --> 00:44:41.110
Applied math comes
in this structure.
00:44:41.110 --> 00:44:43.690
The only thing I haven't
got yet in the picture
00:44:43.690 --> 00:44:49.260
is an outside source to
make something happen.
00:44:49.260 --> 00:44:52.650
I could add a
current source here,
00:44:52.650 --> 00:44:55.610
I could, I could add
external currents
00:44:55.610 --> 00:44:57.370
going in and out of nodes.
00:44:57.370 --> 00:44:59.790
I could add batteries
in the edges.
00:44:59.790 --> 00:45:01.530
Those are two ways.
00:45:01.530 --> 00:45:05.860
If I add batteries in the edges,
they, they come into here.
00:45:05.860 --> 00:45:07.650
Let me add current sources.
00:45:07.650 --> 00:45:13.160
If I add current sources,
those come in here.
00:45:13.160 --> 00:45:16.030
So there's a, there's
where current sources go,
00:45:16.030 --> 00:45:22.160
because the F is a like a
current coming from outside.
00:45:22.160 --> 00:45:25.080
So we have our edges,
we have our graph,
00:45:25.080 --> 00:45:33.370
and then I send one amp into
this node and out of this node
00:45:33.370 --> 00:45:36.970
-- and that gives
me, a right-hand side
00:45:36.970 --> 00:45:38.840
in Kirchoff's Current Law.
00:45:38.840 --> 00:45:41.290
And can I -- to
complete the lecture,
00:45:41.290 --> 00:45:45.420
I'm just going to put these
three equations together.
00:45:45.420 --> 00:45:49.240
So I start with x, my unknown.
00:45:49.240 --> 00:45:51.350
I multiply by A.
00:45:51.350 --> 00:45:53.520
That gives me the
potential differences.
00:45:53.520 --> 00:45:57.100
That was our matrix A that
the whole thing started with.
00:45:57.100 --> 00:45:59.720
I multiply by C.
00:45:59.720 --> 00:46:03.280
Those are the physical
constants in Ohm's Law.
00:46:03.280 --> 00:46:05.500
Now I have y.
00:46:05.500 --> 00:46:12.270
I multiply y by A
transpose, and now I have F.
00:46:12.270 --> 00:46:16.660
So there is the whole thing.
00:46:16.660 --> 00:46:22.870
There's the basic
equation of applied math.
00:46:22.870 --> 00:46:28.700
Coming from these three steps,
in which the last step is
00:46:28.700 --> 00:46:30.140
this balance equation.
00:46:30.140 --> 00:46:33.560
There's always a balance
equation to look for.
00:46:33.560 --> 00:46:34.880
These are the --
00:46:34.880 --> 00:46:37.350
when I say the most basic
equations of applied
00:46:37.350 --> 00:46:37.980
mathematics --
00:46:37.980 --> 00:46:41.500
I should say, in equilibrium.
00:46:41.500 --> 00:46:43.820
Time isn't in this problem.
00:46:43.820 --> 00:46:47.800
I'm not -- and Newton's
Law isn't acting here.
00:46:47.800 --> 00:46:51.230
I'm, I'm looking at the --
equations when everything has
00:46:51.230 --> 00:46:54.110
settled down, how do
the currents distribute
00:46:54.110 --> 00:46:55.780
in the network.
00:46:55.780 --> 00:47:00.480
And of course there are
big codes to solve the --
00:47:00.480 --> 00:47:04.860
this is the basic problem
of numerical linear algebra
00:47:04.860 --> 00:47:09.190
for systems of equations,
because that's how they come.
00:47:09.190 --> 00:47:13.320
And my final question.
00:47:13.320 --> 00:47:18.940
What can you tell me about
this matrix A transpose C A?
00:47:18.940 --> 00:47:21.400
Or even A transpose A?
00:47:21.400 --> 00:47:24.310
I'll just close
with that question.
00:47:24.310 --> 00:47:28.280
What do you know about
the matrix A transpose A?
00:47:28.280 --> 00:47:32.400
It is always symmetric, right.
00:47:32.400 --> 00:47:33.350
OK, thank.
00:47:33.350 --> 00:47:38.590
So I'll see you Wednesday for a
full review of these chapters,
00:47:38.590 --> 00:47:41.230
and Friday you get to tell me.
00:47:41.230 --> 00:47:42.780
Thanks.