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PROFESSOR: OK.

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This video is a
different direction.

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It will be about linear
equations and not differential

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equations.

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A matrix is at the
center of this video

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and it's called the
incidence matrix.

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And that incidence matrix tells
me everything about a graph.

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Now, what do I mean
by the word graph?

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I don't mean a graph
of sine x or cosine x.

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The word graph is
used in another way

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completely for some
edges and some nodes.

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So I have some nodes.

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In this case 1, 2, 3, 4 nodes.

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That's my number n.

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The number m is the number of
edges that connect the nodes.

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So I have edge 1 connecting
those nodes, edge 2, edge 3, 4,

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and 5.

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And I didn't put in an edge 6.

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A complete graph would
have all possible edges,

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but a general graph
can have some edges.

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Some pairs of
nodes are connected

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others are not connected.

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So now I want to create the
matrix that shows me everything

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that's in that picture.

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Then I can work with
the matrix and graphs.

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And their matrices are the
number one application,

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number one model for
so many applications,

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like the world wide web.

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The web might have-- every
website would be a node

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and there would be an
edge between two nodes

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if those websites are linked.

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So the world wide
web is a giant graph.

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Or the telephone company
has a giant graph

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in which the nodes
are the telephones,

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and there is an edge when a
call is made from one phone

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to another, between two phones.

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So, nodes and edges.

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And our brain-- which
is the great problem

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of the 21st century is to
understand the graph that

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represents our brain, the
connections of neurons

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in our thinking-- well, that's
a tougher problem than we'll

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solve today.

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Let me work with that graph
and create the matrix.

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So the matrix has five rows
coming from the five edges.

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Let me take the first edge.

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So the first edge, there's
edge number 1, goes from node 1

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to node 2.

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The nodes correspond to columns.

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So if I want an edge
from node 1 to node 2,

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that edge 1 will go in row 1.

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So edge 1.

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First edge is
connected to row 1.

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So that edge goes from node 1
to node 2, so I put a minus 1

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and a 1.

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And it doesn't
touch nodes 3 and 4.

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That's edge 1.

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That's row 1.

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Now that tells me everything
I see about edge 1.

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Edge 2 goes from 1 to 3.

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So I'll put a minus 1,
a 0, and a 1 in row 2

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because row 2 comes from edge
2 and it goes from 1 to 3.

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Edge 3 will give me
row 3, from 2 to 3.

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So edge 3 giving
me row 3, 2 to 3.

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Edge 4 went from 1 to 4.

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So minus 1, nothing, nothing, 1.

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That tells me that edge 4 is
going from node 1 to node 4.

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And finally, from node 2
to node 4 is the final row.

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Do you see there the graph?

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Everything, all the
information in this picture

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is now captured in that matrix.

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So we can work with the matrix.

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And what does a matrix do?

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It multiplies vectors.

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That's what a matrix
does, it acts on vectors.

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So what happens if I multiply
that matrix by a vector?

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So now let me take
out these edge numbers

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and do a multiplication.

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That matrix has four columns,
it's a 5 by 4 matrix, m by n.

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5 by 4.

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So it multiplies a vector with
four components and those four

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components will come
from the four nodes.

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And maybe they represent
voltages at the nodes.

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Let me think like an electrical
engineer for a moment.

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So if there's my
matrix, I imagine

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I have voltages, v1, v2,
v3, v4, at the nodes.

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So there's a v1 voltage
here, v2, a v3, and a v4,

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and where those voltages'
currents will flow.

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So my unknowns are the
voltages, the four voltages,

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and the five currents.

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That's what the
engineer needs to know.

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So first of all, when I multiply
A times v, what do I get?

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Let me just do that
multiplication.

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So that first row times that
gives me v2 minus v1, right?

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The dot product of the
row with the vector.

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The next one is v3 minus v1.

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Then I have a minus 1 there.

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It's a v3 minus a v2.

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Then I have a minus 1 and a 1.

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I think that's v4 minus v1.

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And finally, this
dot product of that

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will give me a v4 minus v2.

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So what am I seeing here?

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This is now A times v.
I've done a multiplication

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by a vector of voltages.

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And what have I found?

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I found the differences
in voltages,

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the voltage difference
between one end

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of the edge and the other one.

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I have five edges and
now I have five results

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and those are the
voltage differences.

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And what does a
difference in voltage

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do if these are at different
voltages, different potentials?

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Current flows.

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If they're at the same
potential, no current flows,

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right?

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That's the fundamental
driving equation

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of currents from voltages is
the difference in the voltage.

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The difference in the
potential drives the flow.

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And now, how much flow?

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So now I'm looking
for the flows.

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So can I call those
w, for the flows.

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So I have a w2 is the
flow on that edge.

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A w1 is a flow there.

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A w5, a w3, and a w4.

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My pair of unknowns-- and that's
the beauty of this picture--

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is the voltages v1 to v4 four
at the nodes, and the currents,

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the flows, w1 to w5
on the five edges.

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And I've seen that Av gives
me the voltage differences.

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I'm going to briefly,
briefly approach

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the fundamental laws of
flow, of current flow,

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of flow in any network.

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We're talking about the
most basic equation,

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I would almost say, of
applied mathematics.

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Maybe I should say of
discrete applied mathematics.

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By discrete I mean a
graph without derivatives.

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I'm not seeing
derivatives here, I'm just

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seeing matrices and vectors.

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So I have to remember
that incidence matrix,

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A-- let me write it down again.

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Av gave the voltage differences.

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And that's one
part of my picture.

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Another part is what is the
equation that finally brings it

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together?

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That if I have the currents--
so the v's were the voltages.

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Now, there's going to
be an equation involving

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w, the currents.

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This, what I'm
going to write here,

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is going to be really important.

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It's going to be Kirchhoff's
Current Law, KCL.

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And I just emphasized that there
are two Hs in Kirchhoff's name.

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So Kirchhoff's
Current Law says--

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and pay attention-- it says
that the total flow into a node

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equals the flow out.

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We're talking about
equilibrium here.

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So if current is traveling
around my graph, my network,

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and it's a stable equilibrium
here so that flow into node 1

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equals flow out of node 1.

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And let me tell you
what that equation is

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in terms of the matrix A.

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This voltage difference
is involved A

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and, beautifully, the
Kirchhoff's Current Law

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involves A transpose.

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So A transpose now is 4 by 5.

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These are the flows, a
vector with five components

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because I have five edges.

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And Kirchhoff's Current
Law would say that's 0.

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So between A and A transpose,
the incidence matrix

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is leading me to the fundamental
equilibrium condition

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for flow in a network.

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Now, one more law is needed.

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It has to connect voltage
differences to flows,

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potentials to currents.

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Do you know who created that
law in electrical engineering?

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It was Ohm.

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So Ohm's Law, finally,
Ohm's Law is edge by edge

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that the potential difference,
the drop in potential,

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the potential forcing current
is proportional to the current.

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So voltage difference--
let me write it in words.

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Voltage difference--
voltage drop

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I could say-- between the
ends or across a resistor

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is proportional to, and
there is some resistance,

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some physical number
comes in here.

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This is where the material
we're working with comes in.

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In Kirchhoff's Laws, those
laws hold for a network

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before we even say what
the network is made of.

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But now if our network is
made of resistors or pipes

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or whatever we have, then
this will be some conductance.

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So E equal IR, some resistance,
times the flow, times

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the current flow, w.

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So a difference in
v's is some number,

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this is the physical
constant that we

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have to measure in a lab to know
how many ohms our resistor is.

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That equation is on each edge.

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So we have a bunch of
equations and together they

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tell us the four voltages
and the five currents.

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And maybe I'll just make
the main point here.

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The main point is that
this matrix is crucial.

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A is crucial.

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A transpose is crucial.

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A gives voltage differences,
it makes something happen.

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A transpose is the balance law,
the balance or current balance

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at each node.

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And you won't be surprised
that when the whole thing is

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put together and we have
a final equation to solve,

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we end up with A
transpose and A.

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00:15:04,845 --> 00:15:08,660
And that magic
combination, A transpose A,

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is central to graph theory.

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It's called the graph
Laplacian and has

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a name and a fame of its own.

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Thank you.