 So we spend a lot of time and effort trying to determine whether a set of vectors is independent. But why bother? To answer this question, let's take a look at a couple of problems. Suppose v is a set of vectors, and let the coordinates of x with respect to v be a1 through an. Let b1 through bn be another set of coordinates of x with respect to v, where ai is not equal to bi for at least one i. In other words, at least one of these coordinates differs. And prove or disprove that v is independent. So there's quite a bit here, and it will help to review some key ideas. First of all, what do we mean by a set of coordinates? And so remember our coordinates are the coefficients of the linear combination that give us a particular vector. Next, we want to remind ourselves what we can do to prove or disprove that a set of vectors is independent. And that's going to occur depending on whether or not the linear combination equal to zero has any but the trivial solution. So let's take a look at our problem. Now we have two sets of coordinates for the same vector x, and that means x could be written as a linear combination using the a's, or it can be written as a linear combination using the b's. We'll rearrange our terms a little bit, and this gives us a linear combination of the v vectors equal to the zero vector. Now remember the assumption that we made is that ai is not equal to bi for at least one of the i's. So that means at least one of these coefficients is not equal to zero. And so that means we can write the zero vector as a non-trivial linear combination of the vectors in our set, which by our theorem means that the vectors are not independent. And again, a good habit to get into is to always ask yourself, self, what else does this tell me? And because the vectors are not independent, we also know that they don't form a basis. So let's flip the question around. Suppose our set of vectors is not a basis for the span of itself. Let's prove or disprove any vector in our vector space has an infinite number of sets of coordinates with respect to our basis. So again, we have our definition for basis, which depends on whether or not our set of vectors are independent, and we have our theorem on independence, which tells us when a set of vectors will be independent. So let's put those together. Since our set of vectors is not a basis, we know that linear combination equal to zero has at least one solution where at least one of our ci's is not equal to zero. So let's consider some vector in our vector space. The coordinates of that vector will be the coefficients of the linear combination that give us that vector. But I also know the coordinates of the zero vector. I know the coefficients of a linear combination that will give me the zero vector. And so if I add the two vectors together, x plus zero vector will be found by some linear combination. Well, x plus the zero vector is just the vector x. And so I have a linear combination that's equal to x, and the coefficients will give us the coordinates. Since we know that at least one of the ci's is not equal to zero, then at least one coefficient will be different. So this gives us a different set of coordinates for our vector x. And so now I have two sets of coordinates for x. So an important idea in mathematics is anything we can do once, we can do as many times as we want. So we can lather, rinse, repeat. Now that I have a set of coordinates for x, I can add the zero vector and get a new set of coordinates for x. And again, now that I have a set of coordinates for x, I can add the zero vector and get another set of coordinates for x. And in fact, we can generalize this. And so once I know one set of coordinates for x, I can produce as many sets of coordinates for x as I want to. And so any vector x will have an infinite number of sets of coordinates with respect to that set of vectors v. Well, again, we can flip the question. If we have something that's not a basis, then every vector has an infinite number of sets of coordinates. What if we have a basis and we'd like there to be only one set of coordinates? So let's try and prove that. Now a useful proof strategy. If you want to show that something is unique, you want to show that anything else with these same properties has to be the same thing. So suppose the coordinates for x are a1 through an, and that another set of coordinates for x are b1 through bn. Remember, the coordinates are the coefficients of the linear combination that give us the vector. And since they both give us the same vector, then if I use the a's as the coefficients, or if I use the b's as the coefficients, I should get the same thing. Now I'll rearrange these things so that I have a linear combination of the v vectors. And at this point I have linear combination equal to zero. And now is a good time to ask yourself, self, what do I know? And what we know is that a set of vectors forms a basis. And that means these vectors have to be independent. And that means the only way we get linear combination equal to zero is that all of the coefficients have to be equal to zero. So we know that a1 minus b1 has to be zero. So that tells us that a1 and b1 are the same thing. Likewise, a2 minus b2 must be zero. And so that tells us that a2 is equal to b2, and so on down the line. And that means that ai will be equal to bi for all values of i. So if I have one set of coordinates for x and another set of coordinates for x, then I can write the zero vector as a linear combination of the difference between the two coordinates. And since v is a basis, then ai and bi have to be the same for all i, and the two sets of coordinates are in fact the same. If we put these problems together, we obtain the following theorems. Our first problem said that if x had two sets of coordinates with respect to v, then v was not independent and couldn't form a basis. And we could flip that around. If v is not a basis, then any vector x has not only two, but an infinite number of sets of coordinates. And so this gives us a theorem that x will have an infinite number of sets of coordinates with respect to v, if and only if v is not a basis. Conversely, we found that if v was a basis, then every vector x will have a unique set of coordinates. And it works the other way around. If every vector x has a unique set of coordinates, then v is going to be a basis. And so we get the theorem x will have a unique set of coordinates with respect to our set of vectors v, if and only if our set of vectors v is a basis. And this is why it's so important to worry about independence. Independence of our set of vectors guarantees that we have a unique set of coordinates for every vector in our vector space. Thank you.