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Okay, this is lecture 12.
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We've reached 12 lectures and this one is more than the other is about applications of linear
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algebra.
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And I'll confess when I'm giving you examples of the null space and the row space, I create
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a little matrix.
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You probably see that I just invent that matrix as I'm going.
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And I feel a little guilty about it 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, they have a definite structure and anybody who works with them
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gets uses that structure.
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Other reports like this weekend I was at an event with chemistry professors.
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Okay, those guys are row reducing matrices and what matrices are they working with?
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Well, the matrices tell them how much of each element goes into the or each molecule, how
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many molecules of each go into a reaction and what comes out.
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And by row reduction they get a clearer picture of a complicated reaction.
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And this weekend I'm going to the to a sort of birthday party at math work.
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So math works is out root nine in NADIC.
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That's where MATLAB is created.
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It's a very, very successful software, 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 about what I think is the most important model in applied
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math and the discrete version is a graph.
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So can I draw a graph right down the matrix that's associated with it and that's a great
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source of matrices?
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You'll see.
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So a graph is just so a graph to repeat has nodes and edges.
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OK, and I'm going to write down the graph, a graph.
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So I'm just creating a small graph here.
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As I mentioned last time we would be very interested in the graph of all websites or the graph
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of all telephones, I mean, or the graph of all people in the world.
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Here let me take just maybe nodes 1, 2, 3.
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I better put in that edge and maybe an edge to a node 4 and another edge to node 4.
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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 4 in my n equal 4 nodes and the matrix will have m equals a number.
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There'll be a row for every edge.
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So I've got 1, 2, 3, 4, 5 edges.
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So that will be the number of rows.
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And I have to write down the matrix that I want to study.
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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 just do that with an arrow.
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Say from 1 to 2, 1 to 3, 2 to 3, 1 to 4, 3 to 4.
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That just tells me if I have current flowing on these edges, then I know whether it's
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accounted as positive or negative according to 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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My example is going to come, the example of the words that I will use, will be words like
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potential, potential difference, currents.
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In other words, I'm thinking of an electrical network.
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But that's just one possibility.
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My applied mass class builds on this example.
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It could be a hydraulic network.
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So we could be doing flow of water, flow of oil.
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Other examples, this could be a structure like a design for a bridge or a design for
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a buckminster fuller dome or many other possibilities.
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So many.
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So let's take potentials and currents as a basic example and let me create the matrix that
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tells you exactly what the graph tells you.
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So now I'll call it the incidence matrix.
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So let me write it down and you'll see 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 this corresponds to node one, two,
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three, and four, the four columns, the five rows corresponds, the first row corresponds
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to edge one.
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Edge one leaves node one and goes into node two and it doesn't touch three and four.
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Edge two.
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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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Oh, I'm discovering no image.
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I've got to write number that twice here, the edge four, and here's edge five.
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OK?
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All right.
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So edge one, as I said, goes from node one to two.
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Edge two goes from two to three, no two to three.
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So 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 tempted to stop for a moment with those three edges.
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Edge is one, two, three.
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Those form what do you call the little subgraph formed by edge is one, two, and three.
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That's a loop.
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The number of loops and the position of the loops will be crucial.
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OK.
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Actually, here's an interesting point about loops.
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If I look at those rows corresponding to edges one, two, three, and these guys made a loop,
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you want to tell me if I just looked at that much of the matrix, it would be natural for
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me to ask, 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 independent?
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You see 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 that's like a hint here that loops correspond to dependent linearly dependent column, linearly
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dependent rows.
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OK.
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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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Can we?
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Here's the review now.
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There's a matrix that comes from somewhere.
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If it was a big graph, it would be a large matrix, but a lot of zeros, right?
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Because every row only has two non-zero.
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So the number of, it's a very sparse matrix.
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The number of non-zero is exactly two times five.
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It's two M. Every row only has two non-zero.
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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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The graphs, the real graphs from 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 the main questions about matrices.
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So first question, what about the null space?
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So what am I asking, if I ask you for the null space of that matrix, I'm asking you if,
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I'm looking at the columns of the matrix, four columns, and I'm asking you, are those
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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 of the columns, it tells us how to combine columns
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to get zero.
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And is there anything in the null space of this matrix other than just the zero vector?
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In other words, are those four columns independent or dependent?
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OK.
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So that's our question.
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Let me, I don't know if you see the answer, whether there's, so, so I, let's see, I guess
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we could do it properly.
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We could solve Ax equals zero.
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So let me solve Ax equals zero to find the null space.
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OK.
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What's Ax?
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And I put X in here in, in little letters, X1, X2, X3, X4.
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That's, it's got four columns, Ax.
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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, I'll put the answer here.
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Ax equals, what's the first component of Ax?
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Can you take that first row?
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Minus 1, 1, 0, 0, and multiply by the X.
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And of course, you get X2 minus X1.
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The second row, I get X3 minus X2.
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From the third row, I get X3 minus X1.
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From the fourth row, I get X4 minus X1.
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And from the fifth row, I get X4 minus X3.
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And I want to know when is the thing zero?
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That's, that's, this is my equation, Ax equals zero.
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Notice what that matrix A is doing.
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What we, we've created a matrix that computes the differences
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across every edge.
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The difference is in potential.
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Let me, let me even begin to give this interpretation.
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I'm going to think of this vector X, which is X1, X2, X3, X4, as the potentials at the
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nodes.
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So I'm introducing a word, potentials at the node.
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And now if I multiply by A, I get these, I get these five components, X2 minus X1,
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et cetera.
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And what are they?
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They're potential differences.
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That's what A computes.
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If I have potentials at the nodes and I multiply by A, it gives me the potential differences.
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The differences in potential across the edges.
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OK, when are those differences all zero?
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So I'm looking for the null space.
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Of course, if all the X is zero, then I get zero.
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That just tells me, of course, there's zero vectors in the null space.
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But there's more in the null space.
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Those columns are of A are dependent, right?
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Because I can find solutions to that equation.
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Tell me the null space.
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Tell me one vector in the null space.
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So tell me an X, it's got four components, and it makes that thing zero.
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So what the good X to do that?
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1, 1, 1, 1.
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Constant potential.
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If the potentials are constant, then all the potential differences are zero, and that
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X is in the null space.
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What else is in the null space?
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Yeah, let me ask you this always.
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Give me a basis for the null space.
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A basis for the null space will be just that.
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That's a good, that's it.
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That's a basis for the null space.
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The null space is actually one dimensional, and it's the line of all vectors through that
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one.
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So there's a basis for it, and here is the whole null space.
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Any multiple of 1, 1, 1, 1, 1, it's the whole line in four dimensional space.
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Do you see that that's the null space?
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So the dimension of the null space of A is 1, and there's a basis for it, and there's
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everything that's in it.
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Good.
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And what does that mean physically?
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I mean, what does that mean in the application?
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That guy in the null space.
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It means that the potentials can only be determined up to a constant.
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Potential differences are what make current flow.
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That's what makes things happen.
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It sees potential differences that will make something move in our network between node
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2 and node 1.
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Everything will move if all potentials are the same.
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If all potentials are C, C, C, and C, then nothing will move.
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So we have this one parameter.
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This arbitrary constant that raises or drops all the potentials.
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It's like ranking football teams, whatever.
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We have a, there's a constant, or looking at temperatures, you know, there's a flow
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of heat from higher temperature to lower temperature.
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If temperatures are equal, there's no flow, and therefore we can measure temperatures,
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we can measure temperatures by Celsius, or we can start at absolute zero.
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And that arbitrary, it's the same arbitrary constant that was there in calculus.
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In calculus, right, when you took the integral, the indefinite integral, there's
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a plus C, and you had to set a starting point to know what that C was.
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So here what often happens is we fixed one of the potentials like the last one.
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So a typical, a typical thing would be to ground that node to set its potential at zero.
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And if we do that, if we fix that potential, so it's not unknown anymore, then that column
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disappears, and we have three columns, and those three columns are independent.
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So I'll leave the column in there, but we'll remember that grounding a node is the way
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to get it out.
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And grounding a node is the way to setting a potential to zero tells us the base for
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all potentials, then we can compute the others.
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Okay, but now I've talked enough to ask what the rank of the matrix is.
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What's the rank then?
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The rank of the matrix, so we have a 5, 5, 4 matrix.
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We've located its null space, one dimensional.
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How many independent columns do we have?
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What's the rank?
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It's three.
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And the first three columns are actually any three columns will be independent.
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Any three potentials are independent, good variables.
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The fourth potential is not, we need to set, and typically we ground that node.
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Okay, rank is three.
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Rank equals three.
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Okay.
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Let's see, do I want to ask you about the column space?
246
00:19:50.880 --> 00:19:54.680
The column space is all combinations of those columns.
247
00:19:54.680 --> 00:19:58.040
I could say more about it than I will.
248
00:19:58.040 --> 00:20:05.080
Let me go to the null space of A transpose.
249
00:20:05.080 --> 00:20:12.360
Because the equation A transpose Y equals zero is probably the most fundamental equation
250
00:20:12.360 --> 00:20:14.120
of applied mathematics.
251
00:20:14.120 --> 00:20:17.680
Let's talk about that.
252
00:20:17.680 --> 00:20:19.680
That deserves our attention.
253
00:20:19.680 --> 00:20:22.040
A transpose Y equals zero.
254
00:20:22.040 --> 00:20:29.040
Let me put it on here.
255
00:20:29.040 --> 00:20:31.960
Okay.
256
00:20:31.960 --> 00:20:35.960
So A transpose Y equals zero.
257
00:20:35.960 --> 00:20:40.080
So I'm, now I'm finding the null space of A transpose.
258
00:20:40.080 --> 00:20:46.400
Oh, and if I ask you a dimension, you could tell me what it is.
259
00:20:46.400 --> 00:20:51.720
What's the dimension of the null space of A transpose?
260
00:20:51.720 --> 00:20:54.280
We now know enough to answer that question.
261
00:20:54.280 --> 00:21:01.280
What's the general formula for the dimension of the null space of A transpose?
262
00:21:01.280 --> 00:21:05.160
A transpose, let me even write out A transpose.
263
00:21:05.160 --> 00:21:10.520
This A transpose will be M by M, right?
264
00:21:10.520 --> 00:21:12.680
N by M.
265
00:21:12.680 --> 00:21:16.760
In this case, it'll be four by five.
266
00:21:16.760 --> 00:21:19.120
Those columns will become rows.
267
00:21:19.120 --> 00:21:28.080
Minus one, zero, minus one, minus one, zero is now the first row.
268
00:21:28.080 --> 00:21:34.320
The second row of the matrix, one, minus one, and three, zeroes.
269
00:21:34.320 --> 00:21:41.440
The third column now becomes the third row, zero, one, one, zero, minus one.
270
00:21:41.440 --> 00:21:48.440
And the fourth column becomes the fourth row.
271
00:21:48.440 --> 00:21:49.440
Okay, good.
272
00:21:49.440 --> 00:21:51.240
There's A transpose.
273
00:21:51.240 --> 00:21:53.720
That multiplies Y.
274
00:21:53.720 --> 00:22:01.120
Y one, Y two, Y three, Y four, and Y five.
275
00:22:01.120 --> 00:22:04.120
Okay.
276
00:22:04.120 --> 00:22:06.640
Now you've had time to think about this question.
277
00:22:06.640 --> 00:22:10.320
What's the dimension of the null space?
278
00:22:10.320 --> 00:22:14.320
If I set all those, wow.
279
00:22:14.320 --> 00:22:22.560
And usually, sometimes during this semester, I'll drop one of these erasers behind there.
280
00:22:22.560 --> 00:22:23.720
That's a great moment.
281
00:22:23.720 --> 00:22:26.840
There's no recovery.
282
00:22:26.840 --> 00:22:32.600
There's centuries of erasers are back there.
283
00:22:32.600 --> 00:22:34.800
Okay.
284
00:22:34.800 --> 00:22:38.600
Okay.
285
00:22:38.600 --> 00:22:42.760
What's the dimension of the null space?
286
00:22:42.760 --> 00:22:48.840
Give me the general formula first in terms of R and M and N.
287
00:22:48.840 --> 00:22:51.400
This is like crucial.
288
00:22:51.400 --> 00:22:57.840
We struggled to decide what dimension meant.
289
00:22:57.840 --> 00:23:04.480
And then we figured out what it equals for an M by N matrix of rank R.
290
00:23:04.480 --> 00:23:08.680
And the answer was M minus R.
291
00:23:08.680 --> 00:23:17.640
There are M equal five components, M equal five columns of A transpose.
292
00:23:17.640 --> 00:23:24.560
And R of those columns are pivot columns because they'll have R, Pivots, it has rank R, and
293
00:23:24.560 --> 00:23:30.960
M minus R are the three ones now for A transpose.
294
00:23:30.960 --> 00:23:33.920
So that's five minus three.
295
00:23:33.920 --> 00:23:37.960
So that's two.
296
00:23:37.960 --> 00:23:42.240
And I would like to find this null space.
297
00:23:42.240 --> 00:23:44.680
I know it's dimension.
298
00:23:44.680 --> 00:23:48.440
Now I want to find out a basis for it.
299
00:23:48.440 --> 00:23:51.720
And I want to understand what this equation is.
300
00:23:51.720 --> 00:23:55.920
So let me say what A transpose Y actually represents.
301
00:23:55.920 --> 00:23:58.400
Why I'm interested in that equation.
302
00:23:58.400 --> 00:23:59.400
Yeah.
303
00:23:59.400 --> 00:24:07.200
I'll put it down with those older erasers and continue this.
304
00:24:07.200 --> 00:24:10.400
Here's the great picture of applied mathematics.
305
00:24:10.400 --> 00:24:12.600
So let me complete that.
306
00:24:12.600 --> 00:24:20.880
There's a matrix that I'll call C that connects potential differences to currents.
307
00:24:20.880 --> 00:24:25.680
So I'll call these their currents on the edges.
308
00:24:25.680 --> 00:24:31.520
Y1, Y2, Y3, Y4, and Y5.
309
00:24:31.520 --> 00:24:37.160
Those are currents on the edges.
310
00:24:37.160 --> 00:24:44.840
And this relation between current and potential difference is Ohm's law.
311
00:24:44.840 --> 00:24:50.080
This here is Ohm's law.
312
00:24:50.080 --> 00:25:00.120
Ohm's law says that the current on an edge is some number times the potential drop.
313
00:25:00.120 --> 00:25:04.320
And that number is the conductance of the edge, one over the resistance.
314
00:25:04.320 --> 00:25:17.120
This is the old current is the relation of current resistance and change in potential.
315
00:25:17.120 --> 00:25:21.600
So it's a change in potential that makes some current happen.
316
00:25:21.600 --> 00:25:25.280
And it's Ohm's law that says how much current happen.
317
00:25:25.280 --> 00:25:26.280
Okay.
318
00:25:26.280 --> 00:25:37.200
The final step of this framework is the equation A transpose Y equals zero.
319
00:25:37.200 --> 00:25:42.360
And that's what is that thing?
320
00:25:42.360 --> 00:25:43.680
It has a famous name.
321
00:25:43.680 --> 00:25:56.240
It's Kirkoff's current law, KCL, Kirkoff's current law, A transpose Y equals zero.
322
00:25:56.240 --> 00:26:00.920
So that when I'm solving it, when I go back up with this blackboard and solve A transpose
323
00:26:00.920 --> 00:26:12.080
Y equals zero, it's this pattern of that I want you to see.
324
00:26:12.080 --> 00:26:16.560
We had rectangular matrices and real applications.
325
00:26:16.560 --> 00:26:21.760
But in those real applications comes A and A transpose.
326
00:26:21.760 --> 00:26:26.840
So our four sub spaces are exactly the right things to know about.
327
00:26:26.840 --> 00:26:29.920
All right, let's know about that.
328
00:26:29.920 --> 00:26:32.280
No space of A transpose.
329
00:26:32.280 --> 00:26:35.360
Wait a minute, where'd it go?
330
00:26:35.360 --> 00:26:37.040
Here it is.
331
00:26:37.040 --> 00:26:38.040
Okay.
332
00:26:38.040 --> 00:26:39.560
Okay.
333
00:26:39.560 --> 00:26:41.560
No space of A transpose.
334
00:26:41.560 --> 00:26:46.560
We know what its dimension should be.
335
00:26:46.560 --> 00:26:47.560
Let's find out.
336
00:26:47.560 --> 00:26:51.200
Tell me a vector in it.
337
00:26:51.200 --> 00:26:53.400
Tell me now, so what am I asking you?
338
00:26:53.400 --> 00:27:00.720
I'm asking you for five currents that satisfy Kirkoff's current law.
339
00:27:00.720 --> 00:27:03.280
So we better understand what that law says.
340
00:27:03.280 --> 00:27:05.760
That law A transpose Y equals zero.
341
00:27:05.760 --> 00:27:08.760
What does that say?
342
00:27:08.760 --> 00:27:12.920
Say in the first row of A transpose.
343
00:27:12.920 --> 00:27:21.160
That says, so the first row of A transpose says minus Y1 minus Y3 minus Y4.
344
00:27:21.160 --> 00:27:26.200
Y4 is zero.
345
00:27:26.200 --> 00:27:28.480
Where did that equation come from?
346
00:27:28.480 --> 00:27:30.880
Let me redraw the graph.
347
00:27:30.880 --> 00:27:38.680
Can I redraw the graph here so that we, maybe here, so that we see again the, there was
348
00:27:38.680 --> 00:27:46.920
node one, node two, node three, node four was off here.
349
00:27:46.920 --> 00:27:48.800
That was our graph.
350
00:27:48.800 --> 00:27:50.400
We had currents on those.
351
00:27:50.400 --> 00:27:54.120
We had current Y1 going there.
352
00:27:54.120 --> 00:27:57.880
We had current Y, what were the other, what are those edge numbers?
353
00:27:57.880 --> 00:28:05.040
Y4 here and Y3 here.
354
00:28:05.040 --> 00:28:09.160
And then a Y2 and a Y5.
355
00:28:09.160 --> 00:28:14.320
I'm just topping what was on the other board so it's the convenient to see it.
356
00:28:14.320 --> 00:28:17.560
What is this equation telling me?
357
00:28:17.560 --> 00:28:22.880
It's first equation of Kirk-Off's current law.
358
00:28:22.880 --> 00:28:24.880
What does that mean for that graph?
359
00:28:24.880 --> 00:28:32.920
Well, I see Y1, Y3, and Y4 as the current leaving node one.
360
00:28:32.920 --> 00:28:38.360
So sure enough, the first equation refers to node one and what does it say?
361
00:28:38.360 --> 00:28:43.400
It says that the net flow is zero.
362
00:28:43.400 --> 00:28:50.640
That equation A transpose Y, Kirk-Off's current law is a balanced equation, a conservation
363
00:28:50.640 --> 00:28:56.160
law, physicist, be overjoyed right by this stuff.
364
00:28:56.160 --> 00:29:06.080
It says that M equals L. And in this case, the three arrows are all going out, so it says
365
00:29:06.080 --> 00:29:08.840
Y1, Y3, Y4, add to zero.
366
00:29:08.840 --> 00:29:10.920
Let's take the next one.
367
00:29:10.920 --> 00:29:17.680
The second row is Y1 minus Y2.
368
00:29:17.680 --> 00:29:19.880
And that's all that's in that row.
369
00:29:19.880 --> 00:29:23.600
And that must have something to do with node two.
370
00:29:23.600 --> 00:29:29.440
And sure enough, it says Y1 equals Y2, current, the n equals current L.
371
00:29:29.440 --> 00:29:31.200
The third one.
372
00:29:31.200 --> 00:29:38.320
Y2 plus Y3 minus Y5 equals zero.
373
00:29:38.320 --> 00:29:41.880
That certainly would be what's up at the third node.
374
00:29:41.880 --> 00:29:47.120
Y2 coming in, Y3 coming in, Y5 going out, add to balance.
375
00:29:47.120 --> 00:29:58.640
And finally, Y4 plus Y5 equals zero says that at this node, Y4 plus Y5, the total flow
376
00:29:58.640 --> 00:30:01.560
is zero.
377
00:30:01.560 --> 00:30:08.920
We don't charge dozens of cumulade at the node, it travels around.
378
00:30:08.920 --> 00:30:09.920
Okay.
379
00:30:09.920 --> 00:30:16.480
Now, give me, I come back now to the linear algebra question.
380
00:30:16.480 --> 00:30:21.280
What's a vector Y that solves these equations?
381
00:30:21.280 --> 00:30:31.120
Can I figure out what the null space is for this matrix, a transpose, by looking at
382
00:30:31.120 --> 00:30:32.120
the graph?
383
00:30:32.120 --> 00:30:36.840
I'm happy if I don't have to do elimination.
384
00:30:36.840 --> 00:30:38.240
I can do elimination.
385
00:30:38.240 --> 00:30:39.800
We know how to do.
386
00:30:39.800 --> 00:30:43.000
We know how to find a null space basis.
387
00:30:43.000 --> 00:30:47.880
We can do elimination on this matrix.
388
00:30:47.880 --> 00:30:54.160
And we'll get it into a good reduced row echelon form and the special solutions will pop
389
00:30:54.160 --> 00:30:55.440
right out.
390
00:30:55.440 --> 00:31:00.000
But I would like to even do it without that.
391
00:31:00.000 --> 00:31:10.680
Let me just ask you first, if I did elimination on that matrix, what would the last row become?
392
00:31:10.680 --> 00:31:16.880
What would the last row, if I do elimination on that matrix, the last row of R will be
393
00:31:16.880 --> 00:31:20.360
all zeroes, right?
394
00:31:20.360 --> 00:31:21.360
Why?
395
00:31:21.360 --> 00:31:23.720
Because the rank is three.
396
00:31:23.720 --> 00:31:26.920
We only have three pivot.
397
00:31:26.920 --> 00:31:30.520
And the fourth row will be all zeroes when we eliminate.
398
00:31:30.520 --> 00:31:40.280
So elimination will tell us what we spotted earlier, what's the null space, all the information,
399
00:31:40.280 --> 00:31:42.480
what are the dependencies?
400
00:31:42.480 --> 00:31:44.880
We'll find those by elimination.
401
00:31:44.880 --> 00:31:49.040
But here, in a real example, we can find them by thinking.
402
00:31:49.040 --> 00:31:51.040
Okay.
403
00:31:51.040 --> 00:31:56.400
Again, my question is, what is a solution Y?
404
00:31:56.400 --> 00:32:06.560
How could current travel around this network without collecting any charge at the nodes?
405
00:32:06.560 --> 00:32:07.720
Tell me a Y.
406
00:32:07.720 --> 00:32:08.720
Okay.
407
00:32:08.720 --> 00:32:16.720
So the basis for the null space of A transpose.
408
00:32:16.720 --> 00:32:19.320
How many vectors am I looking for?
409
00:32:19.320 --> 00:32:20.320
Two.
410
00:32:20.320 --> 00:32:22.840
It's a two-dimensional space.
411
00:32:22.840 --> 00:32:27.840
My basis should have two vectors in it, give me one.
412
00:32:27.840 --> 00:32:29.400
One set of current.
413
00:32:29.400 --> 00:32:32.640
Suppose, let me start it.
414
00:32:32.640 --> 00:32:35.640
Let me start with Y1 as one.
415
00:32:35.640 --> 00:32:36.640
Okay.
416
00:32:36.640 --> 00:32:44.400
So one unit of one amp travels on edge one with the arrow.
417
00:32:44.400 --> 00:32:45.400
Okay.
418
00:32:45.400 --> 00:32:46.400
Then what?
419
00:32:46.400 --> 00:32:49.160
What is Y2?
420
00:32:49.160 --> 00:32:51.240
It's one also, right?
421
00:32:51.240 --> 00:32:56.840
And of course, what you did was solve Kirchhoff's current law quickly in the second equation.
422
00:32:56.840 --> 00:32:57.840
Okay.
423
00:32:57.840 --> 00:33:05.080
Now we've got one amp leaving node one, coming around to node three, what should we do now?
424
00:33:05.080 --> 00:33:07.760
Well, what shall I take for Y3?
425
00:33:07.760 --> 00:33:15.800
In other words, I've read a choice, but why not make it what you said, negative one?
426
00:33:15.800 --> 00:33:23.480
So I have just sent current one amp around that loop.
427
00:33:23.480 --> 00:33:27.360
What shall Y4 and Y5 be in this case?
428
00:33:27.360 --> 00:33:29.400
We could take them to be zero.
429
00:33:29.400 --> 00:33:35.760
This satisfies Kirchhoff's current law.
430
00:33:35.760 --> 00:33:41.400
We could check it patiently that minus Y1 minus Y3 gives zero.
431
00:33:41.400 --> 00:33:43.200
We know Y1 is Y2.
432
00:33:43.200 --> 00:33:47.200
The other is Y4 plus Y5 is certainly zero.
433
00:33:47.200 --> 00:33:52.680
Any current around the loop satisfies the current law.
434
00:33:52.680 --> 00:33:53.680
Okay.
435
00:33:53.680 --> 00:33:57.560
Now you know how to get another one.
436
00:33:57.560 --> 00:34:00.240
Take current around this loop.
437
00:34:00.240 --> 00:34:09.560
So now let Y3 be one, Y5 be one, and Y4 be minus one.
438
00:34:09.560 --> 00:34:16.120
So we have the first basis vector sent current around that loop.
439
00:34:16.120 --> 00:34:20.840
The second basis vector sends current around that loop.
440
00:34:20.840 --> 00:34:22.640
And those are independent.
441
00:34:22.640 --> 00:34:31.120
And I've got two solutions, two vectors in the null space of A transpose, two solutions
442
00:34:31.120 --> 00:34:33.080
to Kirchhoff's current law.
443
00:34:33.080 --> 00:34:39.280
Of course you would say what about sending current around the big loop?
444
00:34:39.280 --> 00:34:41.520
What about that vector?
445
00:34:41.520 --> 00:34:51.520
One for Y1, one for Y2, nothing on Y3, one for Y5 and minus one for Y4.
446
00:34:51.520 --> 00:34:53.160
What about that?
447
00:34:53.160 --> 00:34:57.240
Is that in the null space of A transpose?
448
00:34:57.240 --> 00:34:59.000
Sure.
449
00:34:59.000 --> 00:35:06.440
So why don't we now have a third vector in the basis?
450
00:35:06.440 --> 00:35:10.440
Because it's not independent, right?
451
00:35:10.440 --> 00:35:12.000
It's not independent.
452
00:35:12.000 --> 00:35:15.400
This vector is the sum of those two.
453
00:35:15.400 --> 00:35:21.200
If I send current around that and around that, then on this edge Y3 it's going to cancel
454
00:35:21.200 --> 00:35:27.720
out and I'll have all together current around the whole, the outside loop.
455
00:35:27.720 --> 00:35:33.640
That's what this one is, but it's a combination of those two.
456
00:35:33.640 --> 00:35:43.360
You see that I've now identified the null space of A transpose, but more than that we've
457
00:35:43.360 --> 00:35:53.160
solved Kirchhoff's current law and understood it in terms of the network.
458
00:35:53.160 --> 00:35:54.160
Okay.
459
00:35:54.160 --> 00:35:57.520
So that's the null space of A transpose.
460
00:35:57.520 --> 00:36:02.720
I guess I, there's always one more space to ask you about.
461
00:36:02.720 --> 00:36:15.200
Let's see, I guess I need the row space of A, the column space of A transpose.
462
00:36:15.200 --> 00:36:18.440
So what's the dimension?
463
00:36:18.440 --> 00:36:19.440
Yep.
464
00:36:19.440 --> 00:36:23.080
What's the dimension of the row space of A?
465
00:36:23.080 --> 00:36:26.440
If I look at the original A, it had five rows.
466
00:36:26.440 --> 00:36:30.160
How many were independent?
467
00:36:30.160 --> 00:36:35.480
I guess I'm asking you the rank again, right?
468
00:36:35.480 --> 00:36:38.240
And the answer is three, right?
469
00:36:38.240 --> 00:36:40.360
Three independent rows.
470
00:36:40.360 --> 00:36:43.520
When I transpose it, there's three independent columns.
471
00:36:43.520 --> 00:36:45.320
Are those columns independent?
472
00:36:45.320 --> 00:36:47.320
Those three?
473
00:36:47.320 --> 00:36:51.560
The first three columns, are they the pivot columns of the matrix?
474
00:36:51.560 --> 00:36:53.960
No.
475
00:36:53.960 --> 00:36:56.200
Those three columns are not independent.
476
00:36:56.200 --> 00:37:00.800
In fact, this tells me a relation between them.
477
00:37:00.800 --> 00:37:06.560
There's a vector in the null space that says the first column plus the second column equals
478
00:37:06.560 --> 00:37:08.200
the third column.
479
00:37:08.200 --> 00:37:12.280
They're not independent because they come from a loop.
480
00:37:12.280 --> 00:37:20.480
So the pivot columns, the pivot columns of this matrix will be the first, the second,
481
00:37:20.480 --> 00:37:23.680
not the third, but the fourth.
482
00:37:23.680 --> 00:37:29.320
One, columns one, two, and four are okay.
483
00:37:29.320 --> 00:37:30.640
Where are they?
484
00:37:30.640 --> 00:37:35.160
Those are the columns of A transpose, those correspond to edges.
485
00:37:35.160 --> 00:37:38.120
So there's edge one.
486
00:37:38.120 --> 00:37:40.880
There's edge two.
487
00:37:40.880 --> 00:37:47.240
And there's edge four.
488
00:37:47.240 --> 00:37:53.640
So there's a, that's like, that is a smaller graph.
489
00:37:53.640 --> 00:38:03.000
If I just look at the part of the graph that I've used with thick edges, it has the same
490
00:38:03.000 --> 00:38:05.120
four nodes.
491
00:38:05.120 --> 00:38:08.880
It only has three edges.
492
00:38:08.880 --> 00:38:13.600
And those edges correspond to the independent guys.
493
00:38:13.600 --> 00:38:20.800
And in the graph, those three edges have no loop, right?
494
00:38:20.800 --> 00:38:23.800
The independent ones are the ones that don't have a loop.
495
00:38:23.800 --> 00:38:27.720
All the dependencies came from loops.
496
00:38:27.720 --> 00:38:30.720
They were the things in the null space of A transpose.
497
00:38:30.720 --> 00:38:35.400
If I take three pivot columns, there are no dependencies among them.
498
00:38:35.400 --> 00:38:38.040
And they form a graph without a loop.
499
00:38:38.040 --> 00:38:42.600
And I just want to ask you what's the name for a graph without a loop.
500
00:38:42.600 --> 00:38:49.440
So a graph without a loop has got not very many edges, right?
501
00:38:49.440 --> 00:38:51.400
I've got four nodes.
502
00:38:51.400 --> 00:38:53.160
And it only has three edges.
503
00:38:53.160 --> 00:38:58.360
And if I put another edge in, I would have a loop.
504
00:38:58.360 --> 00:39:01.920
So it's this graph with no loops.
505
00:39:01.920 --> 00:39:06.640
And it's the one where the rows of A are independent.
506
00:39:06.640 --> 00:39:09.400
And what's a graph called that has no loops?
507
00:39:09.400 --> 00:39:11.760
It's called a tree.
508
00:39:11.760 --> 00:39:22.320
So a tree is the name for a graph with no loops.
509
00:39:22.320 --> 00:39:33.200
And just to take one last step here, using our formula for dimension,
510
00:39:33.200 --> 00:39:35.600
using our formula for dimension.
511
00:39:35.600 --> 00:39:46.080
Let's look once at this formula.
512
00:39:46.080 --> 00:39:55.080
The dimension of the null space of A transpose is m minus r.
513
00:39:55.080 --> 00:39:56.320
OK.
514
00:39:56.320 --> 00:39:58.160
This is the number of loops.
515
00:40:01.160 --> 00:40:03.840
Number of independent loops.
516
00:40:03.840 --> 00:40:06.400
m is the number of edges.
517
00:40:09.360 --> 00:40:10.680
And what is r?
518
00:40:13.920 --> 00:40:15.960
What is r for r?
519
00:40:15.960 --> 00:40:17.520
I have to remember way back.
520
00:40:17.520 --> 00:40:23.160
The rank came from looking at the columns of our matrix.
521
00:40:23.160 --> 00:40:25.080
So what's the rank?
522
00:40:25.080 --> 00:40:30.760
That's just remember rank was, remember,
523
00:40:30.760 --> 00:40:33.320
there was one, we had a one dimensional,
524
00:40:33.320 --> 00:40:34.920
the rank was n minus 1.
525
00:40:34.920 --> 00:40:38.800
That's what I'm struggling to say.
526
00:40:38.800 --> 00:40:43.400
Because there were n columns coming from the n nodes.
527
00:40:43.400 --> 00:40:50.840
So it's minus the number of nodes minus 1.
528
00:40:50.840 --> 00:40:57.080
Because of that C, that 111, 111 vector in the null space,
529
00:40:57.080 --> 00:40:59.400
the columns were not independent.
530
00:40:59.400 --> 00:41:03.440
There was one dependency, so we needed n minus 1.
531
00:41:03.440 --> 00:41:06.840
This is a great formula.
532
00:41:06.840 --> 00:41:14.640
This is like the first, shall I write it slightly differently?
533
00:41:14.640 --> 00:41:23.160
The number of edges, let me put things, have I got it right?
534
00:41:23.160 --> 00:41:25.480
Number of edges is m.
535
00:41:25.480 --> 00:41:27.480
The number is m minus r.
536
00:41:27.480 --> 00:41:31.400
OK, so I'm getting, let me put the number of nodes
537
00:41:31.400 --> 00:41:32.320
on the other side.
538
00:41:32.320 --> 00:41:40.200
So the number of nodes, I'll move that to the other side.
539
00:41:40.200 --> 00:41:50.160
Minus the number of edges plus the number of loops
540
00:41:50.160 --> 00:41:55.400
is minus minus 1 is 1.
541
00:41:55.400 --> 00:41:59.320
The number of nodes minus the number of edges plus the number
542
00:41:59.320 --> 00:42:01.400
of loops is 1.
543
00:42:01.400 --> 00:42:03.800
These are like zero dimensional guys.
544
00:42:03.800 --> 00:42:06.640
They're the points on the graph.
545
00:42:06.640 --> 00:42:09.160
The edges are like one dimensional things.
546
00:42:09.160 --> 00:42:11.840
They connect nodes.
547
00:42:11.840 --> 00:42:14.800
The loops are like two dimensional things.
548
00:42:14.800 --> 00:42:17.640
They have like an area.
549
00:42:17.640 --> 00:42:21.440
And this count works for every graph.
550
00:42:21.440 --> 00:42:28.400
And it's known as Euler's formula.
551
00:42:28.400 --> 00:42:29.760
We see Euler again.
552
00:42:29.760 --> 00:42:32.200
That guy never stopped.
553
00:42:32.200 --> 00:42:34.360
OK.
554
00:42:34.360 --> 00:42:36.760
And can we just check?
555
00:42:36.760 --> 00:42:38.840
So what am I saying?
556
00:42:38.840 --> 00:42:42.800
I'm saying that linear algebra proves Euler's formula.
557
00:42:42.800 --> 00:42:49.200
Euler's formula is this great topology fact about any graph.
558
00:42:49.200 --> 00:42:51.120
I'll draw, let me draw another graph.
559
00:42:51.120 --> 00:42:57.440
Let me draw a graph with more edges and loops.
560
00:42:57.440 --> 00:43:00.040
Let me put in lots of, OK.
561
00:43:00.040 --> 00:43:02.040
I just drew a graph there.
562
00:43:02.040 --> 00:43:04.480
So what are the quantities in that formula?
563
00:43:04.480 --> 00:43:06.360
How many nodes have I got?
564
00:43:06.360 --> 00:43:09.680
Looks like five.
565
00:43:09.680 --> 00:43:11.240
How many edges have I got?
566
00:43:11.240 --> 00:43:16.200
1, 2, 3, 4, 5, 6, 7?
567
00:43:16.200 --> 00:43:17.680
How many loops have I got?
568
00:43:17.680 --> 00:43:25.560
1, 2, 3, and Euler's right, I always get 1.
569
00:43:25.560 --> 00:43:33.280
This formula is extremely useful in understanding
570
00:43:33.280 --> 00:43:36.080
the relation of these quantities.
571
00:43:36.080 --> 00:43:38.040
The number of nodes, the number of edges,
572
00:43:38.040 --> 00:43:40.360
and the number of loops.
573
00:43:40.360 --> 00:43:42.200
OK.
574
00:43:42.200 --> 00:43:47.880
Just complete this lecture by completing this picture, this cycle.
575
00:43:47.880 --> 00:44:03.040
So let me come to the equation of applied math.
576
00:44:03.040 --> 00:44:06.480
Let me call these potential differences, say E.
577
00:44:06.480 --> 00:44:08.720
So E is AX.
578
00:44:08.720 --> 00:44:13.600
That's the equation for this step.
579
00:44:13.600 --> 00:44:20.600
The currents come from the potential differences, Y is CE.
580
00:44:20.600 --> 00:44:25.840
The current satisfy Kirchhoff's current law.
581
00:44:25.840 --> 00:44:30.880
Those are the equations with no source terms.
582
00:44:30.880 --> 00:44:34.280
Those are the equations of electrical circuits.
583
00:44:34.280 --> 00:44:43.560
Those are the most basic three equations.
584
00:44:43.560 --> 00:44:46.800
Applied math comes in this structure.
585
00:44:46.800 --> 00:44:49.360
The only thing I haven't got yet in the picture
586
00:44:49.360 --> 00:44:54.960
is an outside source to make something happen.
587
00:44:54.960 --> 00:44:58.720
I could add a current source here.
588
00:44:58.720 --> 00:45:03.080
I could add external currents going in and out of nodes.
589
00:45:03.080 --> 00:45:05.480
I could add batteries in the edges.
590
00:45:05.480 --> 00:45:07.240
Those are two ways.
591
00:45:07.240 --> 00:45:11.600
If I add batteries in the edges, they come into here.
592
00:45:11.600 --> 00:45:13.400
Let me add current sources.
593
00:45:13.400 --> 00:45:18.920
If I add current sources, those come in here.
594
00:45:18.920 --> 00:45:22.120
So there's where current sources go.
595
00:45:22.120 --> 00:45:27.920
Because the F is like a current coming from outside.
596
00:45:27.920 --> 00:45:29.360
So we have our edges.
597
00:45:29.360 --> 00:45:31.120
We have our graph.
598
00:45:31.120 --> 00:45:39.360
And then I send 1 amp into this node and out of this node.
599
00:45:39.360 --> 00:45:44.600
And that gives me a right hand side in Kirchhoff's current law.
600
00:45:44.600 --> 00:45:47.480
And to complete the lecture, I'm just
601
00:45:47.480 --> 00:45:51.240
going to put these three equations together.
602
00:45:51.240 --> 00:45:55.040
So I start with X, my unknown.
603
00:45:55.040 --> 00:45:59.560
I multiply by A. That gives me the potential differences.
604
00:45:59.560 --> 00:46:03.000
With our matrix A, that's the whole thing started with,
605
00:46:03.000 --> 00:46:09.200
I multiply by C. Those are the physical constants in Ohm's law.
606
00:46:09.200 --> 00:46:15.320
Now I have Y. I multiply Y by A transpose.
607
00:46:15.320 --> 00:46:22.600
And now I have F. So there's the whole thing.
608
00:46:22.600 --> 00:46:28.840
There's the basic equation of applied math.
609
00:46:28.840 --> 00:46:34.400
Coming from these three steps in which the last step
610
00:46:34.400 --> 00:46:36.120
is this balance equation.
611
00:46:36.120 --> 00:46:40.840
There's always a balance equation to look for.
612
00:46:40.840 --> 00:46:43.920
When I say the most basic equations of applied mathematics,
613
00:46:43.920 --> 00:46:47.480
I should say in equilibrium.
614
00:46:47.480 --> 00:46:50.760
Time isn't in this problem.
615
00:46:50.760 --> 00:46:54.640
Newton's law isn't acting here.
616
00:46:54.640 --> 00:46:58.360
I'm looking at the equations when everything is settled down.
617
00:46:58.360 --> 00:47:01.840
How do the currents distribute in the network?
618
00:47:01.840 --> 00:47:06.880
And of course, there are big codes to solve.
619
00:47:06.880 --> 00:47:11.440
This is the basic problem of numerical linear algebra
620
00:47:11.440 --> 00:47:13.280
for systems of equations.
621
00:47:13.280 --> 00:47:15.280
Because that's how they come.
622
00:47:15.280 --> 00:47:20.280
And my final question, what can you tell me
623
00:47:20.280 --> 00:47:27.600
about this matrix A transpose CA, or even A transpose A?
624
00:47:27.600 --> 00:47:30.400
Just close with that question.
625
00:47:30.400 --> 00:47:34.400
What do you know about the matrix A transpose A?
626
00:47:34.400 --> 00:47:37.760
It is always the matrix.
627
00:47:37.760 --> 00:47:38.520
Right.
628
00:47:38.520 --> 00:47:39.480
OK, thanks.
629
00:47:39.480 --> 00:47:44.960
So I'll see if Wednesday for a full review of these chapters
630
00:47:44.960 --> 00:47:47.840
and Friday, you get to tell me.
631
00:47:47.840 --> 00:47:57.840
Hey.