Computing the Four Fundamental Subspaces
BEN HARRIS: Hi, and welcome back. Today we're going to do a problem about the four fundamental subspaces. So here we have a matrix B. B is written as the product of a lower triangular matrix and an upper triangular matrix. And we're going to find a basis for, and compute the dimension of, each of the four fundamental subspaces of B. I'll give you a minute to try that on your own, to hit pause, and then I'll be right back in just a minute and we can do it together. OK. We're back. Now, the first thing to notice is that this is an LU decomposition of B. So we have L here and U here. Now let's go one space at a time. Let's start with the column space. And first, let's just say what the dimension of the column space is. OK, so let's look at our U matrix. How many pivots do we have? We have two pivots, so the column space has dimension 2. This is the number of pivots. Good. Now, how do we find a basis for the column space? How do we find tha...