 So we've been talking about how one can construct algebraic representations of these geometric affine sets Both using the bottom up and the top-down approach Well, what happens if we want to start intersecting these things like after all solving a system of equations Oftentimes comes down to having you know, like if we go to the two-by-two case We have two lines intersect each other and so we want to figure out where's that point of intersection Lines themselves are one flats the intersection of these two lines is a zero flat a.k.a a point So it is very relevant to ask how does one intersect affine sets? How do we compute some way of representing those affine sets? Well, the way you would try to determine the intersection is going to be dependent on how you represent the The affine sets so if you describe your flat using the top-down approach That means you have your flats are described by solution sets to systems of linear equations. You might have like a 11 x 1 plus a 1 2 x 2 all the way up to a Let's see 1 n x n right here. This equals b 1 then you have like a 2 1 x 1 plus a 2 2 x 2 all the way up to a 2 n x n b 2 say that's the first that's the first flat so we take To we take two hyperplanes in in dimensional space and we intersect them together This right here is describing some type of like in minus 2 flat and Suppose we have like another one. What if we have something like? We'll have three equations in this next one. So we'll have like a 3 1 x 1 plus a 3 2 x 2 All the way up to a 3 n x n b 3 And you see how this is gonna work. We're gonna get a 4 1 Whoops 4 1 x 1. We're gonna get some a 5 1 x 1 Continue on continue on this would then terminate with a 4 in x n. We also have a a 5 in x n and This is equal to b 4 b 5 just something like that, right? So this right here is representing some type of n minus 3 flat So something that looks like f into the 3 so if we have the intersection of these things So this first the first solution said it's gonna be intersection of two hyperplanes The second one is the intersection of three hyperplanes if we want to intersect these two flats together Well, guess what the top down approach was designed for intersections By construction we build we build flats by intersecting hyperplanes So if we have an intersection of two hyperplanes and an intersection of three hyperplanes If we want to find the intersection of these two flats, well, guess what we just put all of the equations together And we inters we've lived the solution will then be the intersection of all five hyperplanes together And so if you intersect Flats from the top-down approach you just concatenate the two Linear systems together and then solve the combined linear system the solution set, which is a flat will be the intersection of the flats of the two Will be the intersection of the two individual flats from the beginning So solving solving intersection problems with the top-down approach is fairly straightforward The bottom-up approach takes a little bit more effort very still doable, right? But when we're computing the intersection with the bottom-up approach, right? We have to live more careful. So we might have something like the following x is equal to like some x naught We're gonna have some we're gonna have some numbers like say s v 1 plus t v 2 Maybe like some r v 3 like so and then we have another flat Which might be something like x is equal to some other vector x 1 plus like a u 1 plus b U 2 we'll just leave it something like that Right and which case in this situation if our vector, you know We just we want to figure out what vectors belong to both of them We can basically have that oh vector x here is just a variable after all We're trying to figure out that is we can plug these things together and get something like will x naught plus s v 1 plus T v 2 Plus r v 3 is equal to x 1 plus a u 1 plus b u 2 So we have this vector equation which we can translate that into Parametric equations we want to it's it's essentially a linear system in itself We solve that linear system and then we can use that to find The intersection of these two flats given by the bottom-up approach The thing I have to caution you about though is that we very commonly use symbols like s and t to describe Free variables when we talk about the solution set of a system of equations We have to be careful that when we describe the free variables in one so because these are parameters, right? We want to make sure the parameters in one flat don't coincide with parameters of a different flat So when I talked about the second flat I made sure not to reuse the symbol s and t actually use a new symbol a and b right here Because if you use the same symbol twice s and t you might actually get the false impression that the parameters actually have to be equal to each other And so this is the thing is the bottom-up approach We are representing functions by a parameterization, but that parameterization isn't exactly unique Because the spanners are not unique if I had a vector u I could actually replace it with u1 minus u2 right that wouldn't change the that would change the equation But it wouldn't change the flat the the spanners are not themselves unique But it will change like which coefficient you use the parameters are dependent on the spanner But the flat is independent of the spanners and so we don't want to we don't want to think that the only way that The thing intersects is when the parameters are equal. No, no, no We don't need equality on parameters, but we do want the x's to be equal to each other So this one's a little bit more involved. So let me show you an example of this in this lecture We have now seen two planes that we constructed in our Four so and we've we've saw these in previous videos So take a look at the links to the videos you see right now if you want to see the derivations of those So one of the planes Looked like the following. Let me see if I can describe it to us right here The first plane said that x1 equals 26 plus S don't even look at this part right now So the first one said that x1 equals 26 plus s x2 looks like 3 minus 3s x3 is equal to 13 minus 2s plus t and then x4 was equal to negative 18 minus s So those are the four parametric equations that we got for the first plane This is in fact a plane because there are two free variables two parameters in this description S and t are free parameters. They could be whatever you want. That was the first plane Okay, we're racing a racing a racing. Don't mind me a racing All right now the next one is like ignore this part right here We had in a previous example we described a plane in the following way x1 equals negative 17 plus 4a plus 2b x2 was equal to 6 minus 3a you have that x3 is equal to 29 minus 5 minus 4a minus 4b and then x4 is equal to negative 2a minus 2b Now in the previous video when we did this example, I used the parameters s and t You'll now see that I've deliberately changed the parameters to be a and b so that there's no confusion with That this symbol a is independent of this symbol here as well S and a do not have to be the same thing But the first equation this gives me two different ways of representing the coordinate x1 This one right here gives me two different ways of representing the coordinate x2. This is x3 This is x4 right here So for each of the coordinates x1 x2 x3 x4 x5 x6 at however many there are you set the two different Parameterizations equal to each other this parameterization of x1 is Equated to this parameterization of x1 make sure use different parameter symbols on the two different sides of the equations This we take one parameterization for x2 and set it equal to the other parameterization of x2 Make sure you use different symbols for the parameters on the two sides and then you do that for each of those Now this is this is a system of equations. There are four Equations and we have four variables. We have s t a and b and so we actually have some hope that we could solve this for a unique solution So notice what happens if we move everything to the left hand side and maybe times by negative one if you want to But if we move everything to the left hand side, you're gonna get a 4a plus 2b If think about moving actually this way if you move the s to the other side You're gonna get a negative s and then you add 17 to the other side You get 26 plus 17, which is 43 You get this equation right here doing similar things for the other equations You can move things around you get a negative 3a plus 3s equals negative 3 You're gonna get negative 4a minus 4b plus 2s minus t is equal to negative 42 And then lastly negative 2a minus b plus s is equal to negative 18. It's not too difficult to do so Now if you then put this through our typical Robroduction process we compute the r re f Associated to this linear system. I'm gonna skip over the details if that's okay It's a good opportunity for you to pause the video and try it yourself just to confirm it But if we row reduce this system of equations, we get the following solution a equals 8 b equals 9 s Equals 7 and t equals negative 12 now what I am saying is I'm not saying the vector 8 9 7 negative 12 is the vector on this that that's common to both planes That's not what I'm saying. This gives us the parameters if we want to actually know what the vector is We have to go back to the original representations look at the a representation the ab representation for a moment so this one tells me that x1 equals negative 17 plus 4 times 8 plus 2 times 9 and x2 Equals 6 minus 3 times 8 and x3 equals 29 minus 4 times 8 minus 4 times 9 and x4 equals negative 2 times 8 minus 9 You'll see that if I use the ab representation. I actually don't use st At all and when we compute these things this will take a little bit of arithmetic to do here But like 4 times 8 is 32 2 times 8 is sorry 2 times 9 is 18 minus 17 We should end up with 33 when we're done. We do this for the next one. You get a negative 18 4x3 this turns out to be negative 39 and 4x4 You end up with a negative 25 and so this right here is the vector I claim belongs to both planes this right here is our x 33 minus 18 minus 39 minus 25 This would be the solution that this is the vector that's common to both planes now What if we were to do this again using the st? Representation because again, I'm not saying s equals a and t equals b or anything like that I'm saying if we take a different representation x1 is going to equal 26 plus 7 x2 is going to x2 is going to equal 3 minus 3 times 7 x3 is going to equal negative 13 minus 2s and S of course here is 7 and then we're going to add to it t which is negative 12 You might have a little bit more space here and then x4 is Equal to negative 18 minus 7 and so if you simplify each and every one of these things you're going to see that 26 plus 7 is 33 3 minus 3 times 7 is negative 18 negative 13 minus 2 times 7 minus 12 is negative 39 and negative 18 minus 7 is negative 25 So we see that if you take one parameterization using a's and b's I can find that exact same point using a different parameterization using s and t's and this is the only point right here That lives on both of the two planes Now I do also want to kind of point out that we found this we found the point of intersection It's quite involved for the bottom-up approach There's some simplicity of using the the top-down approach when it comes to intersections But I want to come and talk about it something that's interesting in terms of the geometry What we have now demonstrated is I took two planes and Intersected them only at a point. Can we imagine such a picture? Now as you kind of pause and think about it, you're probably like I can't do it, right? If I intersect two planes I'm always gonna get a line, right? Well, that's because you're thinking three dimensionally You have to think about like Doc Brown right here Marty McFly always have this problem You have to think fourth dimensionally that in three dimensions It's not possible for two planes to intersect at a point You have to at least get a line but in four dimensions There's enough extra space that when we drive towards that mural of Indians We will not hit them but we'll pass through the other side, right? Think fourth dimensionally here It's actually possible in four dimensions for two planes to just kiss each other at a specific point like we saw in this example Interesting things happen when we look at geometries that transcend our usual three dimensional of thinking and they can be hard to Visualize it in our minds But mathematics gives us a way and I should say algebra gives us a way to actually think about higher dimensional geometric things Because the algebra will remain the same as we go to the fourth dimension the fifth dimension the sixth dimension the seventh dimension It doesn't really matter as we start thinking higher dimensionally The algebra still works the analogs still work and we can use those to try to understand How one could actually think fourth dimensionally You