 Okay, let's take this very last look at the null space at least for now. Two things I want to get from this. You know, just do some three more examples and then get one level of deeper understanding of what is going on here. I have this 2 by 5 matrix here and it's all zeros. And so that is going to equal the zero vector. On this side, I'm asking, you know, what is the null space of this? Well, remember, I need a linear combination of this column plus a linear combination of that linear combination, linear combination. So some scalar times this, some scalar times that, some scalar times that. So there better be five of these. X2, X sub 3, X sub 4 and X sub 5. So that better give me, I better have five there so that I could get to this side. And think of it, what, you know, I can multiply anything by zero zero and I add to that anything multiplied by zero zero plus anything times zero zero plus anything. So I'm actually, my null space is the whole of our five, there better be five things there. In my null space, I can think about just this linear combination of the columns of the identity matrix. So there is zero one zero zero zero and I have zero zero one zero zero and I have zero zero zero one zero and zero zero zero zero one. Any linear combination of the C1 plus C sub 2 plus C sub 3 plus C sub 4 plus C sub 5 and that is going to give me, you know, any, this C1, C2, C3, C4 and C5 and that can take on any value. C1 to C5 is five scalars that can take any value and what we can see is each of these being a special, a kind of special case and this is linear combinations of each of these and in this instance this might, you might write here as well as X sub 1, X sub 2, X sub 3, X sub 4, X sub 5 just to end up there with X sub 1, X sub 2. So the point being is it can take any real value leaving me with the null space of being this but look at this, they're actually five special cases here. Why is there five special cases? Because, you know, I have five things that I can, that I can change. Just remember that for now. So I have these five special cases. A linear combination of this will give me the full null space which is the full of this five space. If I were to do row, elementary row operations on this, I'm going to end up with, I'm going to end up with this and in essence this is saying X sub 1 equals minus 2 times X sub 2 because it's X sub 1 plus 2 times X sub 2 equals 0 taking the minus X sub 2 over to the other side. So as soon as I fix this, I'm going to let this, I'm going to, this can be anything at once. As soon as I choose it, X1 becomes fixed. So let's choose one that fixes, that fixes the X sub 1 and so that is the special case. The whole of the null space will be a scalar multiple of that and an R2 that is going to be minus 2 plus 1 is going to be this line in R2 and, you know, my null space is anyway, my complete null space is on any way on that line and that is my one special case. Why is there one special case? This is 0, 0 here, nothing to help us there. But here is 1 is 3 and then immediately the other one is fixed. Therefore one special case. Let's do this one. If I were to do this one, I'm going to get 2. So that's just twice 2 of those. I'm going to get this. So what's happening here? I'm saying X sub 1 equals minus twice X sub 2, minus X sub 3, minus twice X sub 4. So how many special cases am I going to get? I'm going to get 3 special cases because 3 of these can vary at any one time and it immediately fixes the other one. I can let these 3 roam 3 and then immediately it fixes this one. So I can let this equal 1 and the other 2 zeros. That means this one is negative 2 for a special solution. I can let this be a 0 and this be a 1 and that be a 0. And that gives me a minus 1 there. There's my second special case and I can have a 0 and a 0 and a 1 there. It fixes that one to minus 2. There's my 3 cases. Why 3 special solutions? Because 3 of these could roam 3 fixing the other one. And now it's just a linear combination of these to give me the full null space. So it's going to be this again, this hyperplane in R4. It's going to be the null space of this matrix of coefficients. Remember this one from before? It had 2 special cases. Why would there be 2? Because I have 1, 2, 3 that can roam 3. But if you think about that this is a 0 and these 2 are linked here. That X sub 3 equals minus X sub 4. That takes away one of the possible 3 that we had and that is going to give us the fact that there's only 2. So you've learned something new by doing new examples but one deeper level of understanding is figuring out how many special cases you are going to get given your matrix of coefficients and how to lead from that to normal examples, how to do a linear combination of those to give us the total null space of our matrix of coefficients that we were given. So that's it for null spaces.