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NIKOLA KAMBUROV: Hi, guys. Today we're going to see how one can use linear algebra to describe graphs and networks. In particular, we'll do the following problem. We're given this very simple graph here with five nodes and six edges. We've already labeled them, and we've put directions on the edges. And we are asked to write down the incidence matrix A, and then to compute its kernel and the kernel of A transpose. And finally, we're asked to compute the trace of A transpose A. I'll give you a few moments to try the problem on your own. And then you'll see my take on it. Hello again. OK, so let's first recall what an incidence matrix is. So an incidence matrix is supposed to encode how the nodes connect to the edges. In particular, it has as many rows as there are edges and as many columns as there are nodes. And we're going to fill in the rows. And we'll fill them out as follows. So we're going to use only negative 1, 1, and...

NIKOLA KAMBUROV: Hi, guys. Today we're going to see how one can use linear algebra to describe graphs and networks. In particular, we'll do the following problem. We're given this very simple graph here with five nodes and six edges. We've already labeled them, and we've put directions on the edges. And we are asked to write down the incidence matrix A, and then to compute its kernel and the kernel of A transpose. And finally, we're asked to compute the trace of A transpose A. I'll give you a few moments to try the problem on your own. And then you'll see my take on it. Hello again. OK, so let's first recall what an incidence matrix is. So an incidence matrix is supposed to encode how the nodes connect to the edges. In particular, it has as many rows as there are edges and as many columns as there are nodes. And we're going to fill in the rows. And we'll fill them out as follows. So we're going to use only negative 1, 1, and...