 So let's talk about row and column spaces, and this emerges as follows. Because I can view a matrix as a collection of vectors, then every matrix defines two vector spaces. First of all, there is a column space, written column A, where I view the columns of the matrix as vectors in a vector space. Now, you should convince yourself that you do actually get a vector space here because our vectors will have to satisfy all those requirements. The other possibility is I might take the row space, written row of A, where I view the diagonal. Wait a minute, no, that would make sense. That doesn't make sense. How about the rows of the matrix as vectors in our vector space? So our column space, we're viewing the columns as vectors. In the row space, we're viewing the rows as vectors. And we'll designate the collection of these vectors using, with a fit of creativity, this script V. And so this is the collection of vectors, either the columns of the matrix or the rows of the matrix. And vector spaces are closed under vector addition and scalar multiplication. And so we're going to define the span of the, the span of the set of vectors to be the set of all linear combinations of the vectors in our collection. So for example, let's take a look at what the column space of this matrix is going to be. So I have a matrix and I have a bunch of columns here. I want to find the column space. I'm going to treat each column as a vector. And I'm going to look at the set of all possible linear combinations of those columns. So linear combinations, I'm going to take each vector and multiply it by some scalar, and I'll add those together. And there's my column space. So here, my set of all possible linear combinations of those four columns. Now we can also introduce something called a subspace. And that is a vector space that is included inside of another vector space. So if I imagine the columns or rows of a matrix forming a vector space, then some of those columns or rows might form a subspace. So for example, let's take our matrix M again. And I can talk about the row space. I'm going to talk about the vector space formed by taking linear combinations of the rows of M. But maybe I could talk about the subspace where I'm only going to use the second row. And so this is going to consist of all linear combinations of just the second row only, which means I can really only look at scalar multiples of that. So note that since we're looking at the row space, I'll go ahead and treat my vectors as being horizontally oriented. So this is linear combination scalar multiples of the vector negative 1, negative 2, 1, and 0. As a general rule, if I'm going to write the vectors horizontally, it's better to separate the entries with a comma. So I might write my subspace this way.