 We've seen how we can use a matrix representation to determine when the linear transformation is one to one. Can we do the same thing for onto? The answer is, yep, we can. What does it mean for a transformation to be onto, right? So suppose we have a linear transformation T, which goes from FN to FM. We say that it's onto if we pick any vector in the target space, the so-called co-domain. If we pick any vector in the target space, call it B, then we can find at least one vector in the domain that maps to it, all right? That's the same thing as saying that the image of T is equal to F of M. So everything in the target space is something we can actually hit, all right? So a transformation being onto actually has something to do with linear systems being consistent. And so what does this mean when we switch to the matrix representation? Imagine the transformation T can be represented by the matrix A, then we can see that the image of T, that is the vectors that come out of our machine, those will coincide with the column space of A, those vectors which can be spanned as linear combinations of the columns of A. The image of a transformation is equal to the column space of its matrix representation. Therefore, if we wanna figure out whether a map is onto or not, because if the map is onto, that means the image should be all of the target space, F M. We just have to show that column space of A is F M. That is, we need to show that the column space is M dimensional. That's what it comes down to. We need to make sure we have M mini-pivots. Let's take a look at some examples right here. So let's take the transformation T, which goes from R2 to R3. It's gonna be given by the formula T of XY is equal to X plus Y comma zero comma two X plus three Y. If we wrote this as a column vector, this would look like X plus Y, first coordinate zero, and then two X plus three Y as the second coordinate, a third coordinate, excuse me. Then we can very quickly see its matrix representation. We're gonna get one, zero, two for the first column, and then we get one, zero, three for the second column. Let's row reduce this thing to echelon form to see where its pivots are gonna be. So we have a pivot in the one, one position. You wanna put rows of zeros at the bottom, so switch those. So we have still this pivot here. We're gonna get rid of the two. We're gonna take row two minus two times row one. We get a minus two, minus two. That then gives us a zero one in the next row. Admittedly, this is an echelon form, so we have enough information to answer the question, but if you wanna continue to row reduce echelon form, you'll take row one minus two times row two. That's how you get a minus two right there. And so now we see the row reduced echelon form of our matrix right here. So what are some things we learned? Well, we have a pivot in every single column, right? We can see the pivot. There's a pivot in the first column. There's gonna be a pivot in the second column. So this tells us that the rank of the matrix A is gonna equal two. So this, so then coming from that, the rank of a matrix is the dimension of the null space, of the column space, excuse me. So we see that the dimension of the column space of A is gonna equal two. On the other hand, what's the dimension of the target space? The target space was R three. The dimension of R three is obviously three. You'll see here there's a disconnect. So the image of this transformation is not R three. So therefore we can see that, oh no, this matrix or this transformation is not onto. So T is not onto. It's not onto because its rank was strictly less than strictly less than the dimension, which in this case was three, all right? This is comparable to what we saw with linear transformations that a matrix, I should say one-to-one linear transformation, so it'll be one-to-one if the nullity is zero. If the nullity is zero, it's one-to-one. If the nullity is greater than zero, it's not one-to-one. That's all you have to do to check whether a transformation is one-to-one or not. When it comes to onto, you look at the rank. If the rank is full, then it'll be onto. If there's a deficit in the rank, then it's not onto. Let's take a look at another example. Let's take S of R three and as it maps onto R two, let's use the formula. S of X, Y, Z equals X plus Y minus two Z and then you get minus Y plus Z right here. Okay, so if we think of this as a column vector, this would look like X plus Y minus two Z and then we get a negative Y plus Z. When you write it this way, you can very quickly see the matrix representation looking at the columns of X's. You're gonna get one zero. Looking at the Y's, you get one and negative one and then looking at the Z's, you're gonna get negative two and one that you see right there. So now let's row reduce this thing. You look at the first pivot position, we're already good to go. Look at the second pivot position, I want that to be a one. So I'm gonna multiply the second row by a negative one and I should admit that this right here is already an echelon form. So in terms of rank and nullity, we actually can already determine what that is but just for the sake of calculation, let's go through it right here. Put in row reduce echelon form. We gotta get rid of this one. So we're gonna take row one minus row two and so then we get the REF right here. So what we see about this transformation, we can see that the rank is the number of pivot columns, the rank is gonna equal two. Not that we need it in this situation because we're trying to compute the, whether it's onto or not, but the nullity is gonna be one, the number of non pivot columns. So this tells us that this matrix right here was not, it's not one to one because the null space is non-trivial, the kernel will be non-trivial. But let's focus on onto. We're trying to map onto the space R2. We're trying to map onto R2. So we want the image, we want the image of S here. Well, this is definitely equal to the column space of the matrix representation A and since the rank is equal to two, this is actually gonna equal the whole thing, R2 right here. So in fact, because the rank is full, it's equal to the dimension of the target space. We can see that absolutely S is in fact onto. And it's a much easier question that to do this with matrices than to do without matrices. We can determine that this map is onto by looking at its rank and therefore you can pick whatever you want. You can pick any vector in R2. Take for example, the vector, let's just do something random, 17 and five, right? Why not? I could find a vector that'll map S via S onto 17.5. Absolutely, because it's onto, that vector is always possible. So summarize what we saw right here. The inconsistency of the equation AX equals B, it may occur when A has a row of zeros and echelon form, but the corresponding position and the augmented column is non-zero. So it's like when you have a matrix, you have like zero, zero, zero, and then you have something not equal to zero over here. This would show you have a contradiction and therefore the system is inconsistent. So in the case of consistency, consistency is dependent on the choice of B and at least one choice of B will have AX equals B inconsistent. So basically in echelon form, if you ever get a row of zeros, that means you can find some B, some B that when you row reduce it, you're gonna get something non-zero here. So if A has a row of zeros in its echelon form, it won't be, the corresponding transmission won't be onto. You can find some vector B that it'll miss. So remember that the rank counts the number of pivot columns. The nullity counts the number of non-pivot columns. The co-rank, remember the co-rank right here, this counts the number of pivot rows, which admittedly the number of pivot rows is the same thing as the number of pivot columns. We're gonna introduce another quantity by analog here. We're gonna call it the co-nullity. Be cautious before you try to use this word in Scrabble. It certainly sounds made up. The co-nullity is gonna be the complement to the nullity. The nullity counts the number of non-pivot columns. So the nullity is gonna count the number of non-pivot, the non-pivot rows. And so the nullity is gonna be the number of rows of zeros in the echelon form of the matrix. If a matrix has a co-nullity greater than one, it's not onto. If it has a nullity greater than, I should say, if the co-nullity is one or greater, it's not onto. If the nullity is greater than equal to one, then it's not one to one. Let's kind of summarize that right here. If the nullity here is greater than equal to one, is greater than equal to one, then we can see that it is not one to one. And in fact, if the nullity is equal to zero, that is when it's one to one. In terms of onto here, we get the following. If the co-nullity is greater than equal to one, then your transformation will not be onto. On the other hand, if the co-nullity, if the co-nullity is equal to zero, then in fact, the transformation will be onto. And so I wanna mention to end this lecture, a very special case. Suppose we have a transformation T, which goes from FN to FM. If M is bigger than N, N is the number of columns, M is the number of rows. So if you have too many rows, if you have too many rows, this suggests, you see right here, T has more rows than columns, then T can't be onto. If you have too many rows here, then T cannot be onto. Because if you have too many rows, then eventually when you reduce this thing, you're gonna have to get a row of zeros. That implies that the co-nullity is positive and so you can't be onto. But on the other hand, what if M is less than N? If M is less than N, that means you have too many columns. Oh boy, too many columns. That is, you have more columns than rows. And so if you have too many columns, that means there will have to be one column without a pivot and your nullity will be positive and therefore you cannot be onto. So we can actually detect these things pretty quickly. If we go back and look at these examples. So let's look at this map right here. You go from two to three right here. This means that your matrix is gonna be three by two. Notice you have too many rows. Too many rows means it's not onto like we observed. Now I cannot say that this map is necessarily one to one just by this information. I'd have to know the pivots, which we saw previously that this one was one to one. On the other hand, if we look at this example right here as we went from R three to R two, the standard matrix representation will be two by three. So it has too many columns. Too many columns suggest that it's gonna be not one to one like we saw in a previous video. But is it onto? It's hard to say we'd have to investigate further. Like we saw on this example, it was in fact onto, but we could construct a map from R three to R two. That's neither one to one nor onto. The not one to one was guaranteed. The not onto, it depends on the matrix. So some of these things we can pick out really quickly because we have too many rows or too many columns. This is why square matrices are the sweet spot. If the Goldilocks of linear algebra, not too hot, not too cold, not too many rows, not too many columns, those are the functions which could be bijective. That is both one to one and onto.