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MIT18_06S10_L12 - 18.06 Linear Algebra Spring 2010...

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18.06 Linear Algebra, Spring 2010 Transcript – Lecture 12 OK. This is lecture twelve. We've reached twelve lectures. And this one is more than the others about applications of linear algebra. And I'll confess. When I'm giving you examples of the null space and the row space, I create a little matrix. You probably see that I just invent that matrix as I'm going. And I feel a little guilty about it, because the truth is that real linear algebra uses matrices that come from somewhere. They're not just, like, randomly invented by the instructor. They come from applications. They have a definite structure. And anybody who works with them gets, uses that structure. I'll just report, like, this weekend I was at an event with chemistry professors. OK, those guys are row reducing matrices, and what matrices are they working with? Well, their little matrices tell them how much of each element goes into the -- or each molecule, how many molecules of each go into a reaction and what comes out. And by row reduction they get a clearer picture of a complicated reaction. And this weekend I'm going to -- to a sort of birthday party at Mathworks. So Mathworks is out Route 9 in Natick. That's where Matlab is created. It's a very, very successful, software, tremendously successful. And the conference will be about how linear algebra is used. And so I feel better today to talk about what I think is the most important model in applied math. And the discrete version is a graph. So can I draw a graph? Write down the matrix that's associated with it, and that's a great source of matrices. You'll see. So a graph is just, so a graph -- to repeat -- has nodes and edges. OK. And I'm going to write down the graph, a graph, so I'm just creating a small graph here. As I mentioned last time, we would be very interested in the graph of all, websites. Or the graph of all telephones. I mean -- or the graph of all people in the world. Here let me take just, maybe nodes one two three -- well, I better put in an -- I'll put in that edge and maybe an edge to, to a node four, and another edge to node four. How's that? So there's a graph with four nodes. So n will be four in my -- n equal four nodes.
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And the matrix will have m equal the number -- there'll be a row for every edge, so I've got one two three four five edges. So that will be the number of rows. And I have to to write down the matrix that I want to, I want to study, I need to give a direction to every edge, so I know a plus and a minus direction. So I'll just do that with an arrow. Say from one to two, one to three, two to three, one to four, three to four. That just tells me, if I have current flowing on these edges then I know whether it's - - to count it as positive or negative according as whether it's with the arrow or against the arrow. But I just drew those arrows arbitrarily. OK.
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MIT18_06S10_L12 - 18.06 Linear Algebra Spring 2010...

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