Hello. And uh today we're gonna be solving a problem in linear algebra up. Excuse me. So let's get started here. So we've been given a factor space P. Of all the polling um meals a degree and over the reels. Um And it's also beginning to subsets of P. Ah They were stated to be the even and er palma meals and people. And those polling males are defined by these two qualities here. And as you can see these are definitions of even in odd functions. Um So one thing to point out here as you might see his little bracket with in the definition of even polynomial. Um And they can ignore this here. Just trying to be fancy if you right you write this this is true for any polynomial for all X. Products and the reels. It's it's true. Um So we don't really need to worry about that bracket. We know this is true right here. So don't worry about that. Um Those who have been given, we've been given the vector space The polynomial degree and we've given the two subsets. One of the even Paul Mills one of the Odd. Now we're asked to prove that even in the odd palma meals are sub spaces of people. So to do that we just have to look at with sub spaces and the definition. And then just prove that that holds I'm and uh subspace has to be closed under addition and scalar multiplication as pretty much the definition of subspace. It's just subset that is closed under addition and scalar multiplication. So that means that if you have any two elements in the subspace or is the kind of subspace, you can multiply them by a real number and you'll also get another element in the same subspace. Or you could add any two elements together. Um In subspace we'll get another element in subspace. So you're stuck. You can't get out you with addition and scale modification. You can't leave. That's basically what it's saying. Um So how do we prove that? Well we just 20 years that we've been given. It's always good with these problems to look back at your the definitions or what they've given you. Um And of course they use some some term like subspace. You could always look up, make sure you know exactly what that means. But we're going to use what they have given us. Let's just take two functions F. N. G. Um So let's say F and G. If you add them together, you get a judge, F N. G. Or even paul no meals. Um Now let's try to prove that that H. Of X is also an even polynomial. Um As you can see if we do this, then we have actually proved that that the even poll numbers are closed under vector edition. Because we added to even polynomial. And we got another one. Okay so let's try to prove that prove that we just use the definition they gave us and work from there. All right. So you see what I did there? I just took it's negative X. Redefined it as we did here. Um And then I redefined these terms using the fact that they're even paul no meals. You can write them also in like this. Um Because H. R. Because we have negative X. Is gonna equal Activex if it's even. And then we're right back where we started and we did 80 bucks. So um That's good to remember. We we didn't have to work too hard for this but we just used definitions. So it was a good thing to remember when you're doing these kind of problems. Now let's do scalar multiplication. We write this little more clearly as you can see. You want to make sure when you're proving something that you define everything pretty clearly to even be constant or define that. Right? So we just did basically the same thing we did before. Um Just redefine get back, get back to where we to find it before appeared. You can read it like that online knowing that Fx says even we can rewrite F. Negative except for X. And and right back where we started. So now we know that even palmas are closed under vector addition and scalar multiplication. Therefore even um even subset upon our meals is also a subspace of P. I am. All right. So let's just anything with the odds. I'm just start doing this because it's pretty much the same thing as before. Birds can apply the odd rule instead. All right. So I hope you so I just did there the same thing we did before. Did you redefine your, we started as what we we stated it was the ancient negative X. Is a combination in summation of F N. G. So we just put that back and and we use the rule for odd Paula. No meals were given before and then we just see the associative property. And um we factored out negative side basically. And yeah, we have shown that um that H of negative ax equals negative H. Of X. The do you sit there and scale of implication here might seem tedious, but it's yes we got to do when you have to prove something can be thorough. So this is definitely using the associative property. There you go. Mm. So we have shown that adding to odd Paula, no meals. We get another Ottawa polynomial. Because this equality is true. And we've all shown that scaling. Yeah polynomial. You will also get there are polynomial because this quality true. There we go. We've showing that both even and odd polynomial. Czar closed under vector addition and scalar multiplication. Therefore God and even Poland is our subspace is of pete. So now we have their second for our question and they are the complementary to subspace complementary. Well we're given well I've provided a definition of what that means. Treaty said basically be complementary. So one is that their intersection is just zero vector. And the other thing is that if you add the two set of spaces together, you will get entire vector space they came from. So another way to think about it is a linear combination. Further. Any element to be in the vector space can be found. There can be reached through a linear combination of now, then one to face the element of the other subspace through. A complimentary. Better way to say is if you add the two subspace together, you get the entire vector space. I think that's the best way to say it. So let's prove it now. So the first thing you want to do, we have taken intersection to even the odd functions and see what we get. Um Well this is actually kind of interesting. But if you want to have an even an odd function, just go and look at our definitions. We just need a function that satisfies both. Use at the same time, let's see if we can just write it out and see if that works. Like if we can think of anything. All right. So because it turns out this is what equality we get. We want to have all of them. If you want to have both those equalities satisfied and have to satisfy this equality here, the polynomial would. Um Well, if you think about it, the only poll numbers that can satisfy All three of these conditions is zero polynomial. Um if you multiply negative one for anything, you're not gonna get the same thing back there no longer to be equal that right away. Should give you a hint that zero is being the shelling answer. Um And that's that's another way you can look at it too. Sleeping it Dimitri frogs around the X equals Y axis. But but for the evens that's that's around the Y axis. So the only you really function that is symmetric around both. That's actually just zero function. So there we go. We've shown that The intersection of the even the Odd Palma Mills is in fact zero. And if you do do some research, he'll realise that zero is even and odd. It's the only even and odd function. I think that's what it could be. Even believe in our polynomial. We're gonna put paul and we're now because because in the vector space, asking the polynomial inherits that from the vector space. So 2nd thing must prove that every polynomial mp can be made up of an even an odd vulnerable. Okay. Um well to prove this, we just have noted polynomial means it's basically the summation of one or more mono meals, whatever I know meals there are constants times variables, two uh to non negative powers. So that's that's that's the definition of polynomial non negative. The key word here is also negative into german hands. So knowing this definition, we can just take a polynomial. I'm gonna just use an example. Just help explain myself a little better but take any polynomial. Um Every polynomial is going to have um Variable terms that you might think. Well what about plus 1? Well this trick here Plus one does have a variable time. Um At least you can write it that way if you want. Uh So now what do you think? What you might ask? Why is this important to know? We're in definitions. This doesn't this is math. Come on. Well, okay so logically speaking every term, every Manno meal and a polynomial must have either an even power for an odd power. Um And you can think about and yeah it's true. Every minimal and a polynomial must have even or an odd power. And what does this mean for us? So, well it means that we can take any polynomial And split it into two. You split it into other polynomial and they if you add them together they will get the original. So you can rewrite any polynomial as a sum of two is a sum of an even and odd polynomial. And does that sound familiar? It should because that is our definition here for any vector in V, w and Z and W W prime respectively. Out together you get the well, same thing. We're all pulling on meals. You can find an even and odd polynomial just by breaking the terms apart. Um If you have them together you will get the polynomial mystery for every polynomial. Just because of this definition. And real quick you might wonder like what about like X squared? There's only even term there. Well, trick question. There's never there's never real. There's a there's always a little trick going on some of these things. But anyway, so for ones like this, you could just say they have experts even and the zero factor is the odd. Um Then yeah, that would be your case where you still haven't uh even an odd function added together to get a polynomial. Uh M. P. I'm saying functions but I met even and odd Paula mill to try to be clear. All right. So, we've proven this. Remember? It didn't ask as they proved that was one point. Didn't say prove. You don't have to be as rigorous as what we did up here, but still keep it. You know, we want to make it logical still. Um But let's go to the last question. So these sub spaces in varying under the differential transformation. So, there are some words we might have to understand here. Uh So differential transformation was that that is a differentiate. Yeah, assuming everyone here knows what should be what differentiating is. Um Yeah, that's differentiating his team derivative. Um It's called the transformation here because as you can see, we took and polynomial where we took it as a technically a polynomial, but there's a polynomial and we took it and we got another polynomial back. So it is a linear transport. Okay, then I could define that, but that's a little out of scope of this. This problem. We already know just from what they told us that the differential transformation is the linear transformation uh because they're they're using it in this context with the vector space. Um But if you didn't want to go prove that, you can go look another video on what differential transformation or where the linear transformation is. You'll see what I mean by that this still in your transformation, but that doesn't really matter for this. We just know from the problem, but this is linear transformation. Um But we want to show that the subspace is in variant um under this transformation. And then you know what that means. So basically I've given the definition here. Um But you can summarize this by saying that if you take an element of a subspace and you apply the transformation, you will also get you'll get back an element of the subspace. So using that transformation you will not leave the subspace. So like we talked about earlier player was being closed under better addition and scalar multiplication. Um And for this problem we we got lucky. It's pretty easy because as you can see what I just did, I just did. We took an even polynomial and we use the differential transformation. And we actually got in Ottawa back. So we left the subspace, we left E. He was left we jumped out so to speak. So is not is not variant under the differential transformation about odds. And I guess they're not going to be either. It didn't pass three there. You see you took there we took a odd all no meal and we got back even. They're both are not in variant right? So you have any questions or any anything you wanted to clarify especially with differential transformation. What that is in terms of like what a linear transformation is. This is a differential transformation taking the derivative. But you don't know what linear transformation is. I'd encourage you just look at other videos on that on that topic. Um Anything else in here? But I hope you were able to see how these problems are able to be solved and hope we learn something. So thank you.