 Once we have a vector space, it makes sense to talk about spanning vectors. So the good news about working in the Latin alphabet is you only have to memorize the shapes of 26 randomly drawn characters to be able to read and write anything. The bad news of working in the Latin alphabet is you only have 26 randomly shaped letters to represent everything. And that means we're going to have to recycle some letters, and we have to distinguish between the different things through various typographical special effects, which are not apparent when they're being read. So here goes. Suppose V is a vector space, and V, the set V1, V2, and so on up to Vn, is a set of vectors in V. If a vector V in V can be expressed as a linear combination of these vectors, then we say that V is in the span of V. More generally, let V be any set of vectors. The set of all linear combinations of these vectors forms the span of V. And it's designated with the rather exotic notation, span V. Let's see if we can prove or disprove that a given set of vectors spans a particular vector space. How about R3? So suppose I have some vector B with components B1, B2, and B3. If possible, I'd like to express my vector B as a linear combination of the vectors in V. And this gives us a system of equations. We can row-reduce the augmented coefficient matrix, then use back substitution to solve for X3, X2, and X1. And notice that this means that whatever the values of B1, B2, and B3 are, we can always find a linear combination that will give us the vector B. In this case, notice that the vectors in our set also lived in R3. And so in this case, we had a set of vectors that spanned all of the vector space they lived in. This doesn't always happen. It's possible that given a set of vectors, they might not span all of the vector space they live in. But whether or not they do, a very important thing happens. If I have any set of vectors which live in some vector space, the span of that set of vectors is also a vector space. And of course, this is something you should prove by showing that the set of linear combinations of these vectors satisfies all of the requirements for getting into club vector space. The vector space that we produce is called a subspace of the original. However, it's important to note that this is an inclusive definition, so the subspace might potentially be the entire thing. And this leads naturally to a new problem, given a set of vectors V, find the subspace they span. And we can solve this problem by solving a system of equations. In this case, we want to find all vectors V for which the equation vector V is a linear combination of our vectors is a solvable equation. Let's take an example. So suppose I have my set of vectors, and I want to describe the vector space that spanned by this set. So I want to find all vectors B that can be expressed as a linear combination of the vectors in our set. So that means our vector is going to be some scalar multiple of one zero one plus some scalar multiple of zero one one. And once we have it in this form, we can rewrite this system component wise. We can reduce the augmented coefficient matrix. And we note the last line translates into the equation zero equals B3 minus B1 minus B2. So that means the vector B1 B2 B3 can only be expressed as a linear combination of one zero one and zero one one, if zero is equal to B3 minus B1 minus B2. And this gives us the vector space spanned by V. Thank you.