25 - Direct sums of subspaces

okay so we want to extend the notion of a sum of two vector spaces or subspaces and define something called a direct sum and here's the motivation I'm going to do a couple more examples of sums and they're going to lead us to this definition let's recall what the definition of a sum is so we have we have two subsets and we're thinking of two subspaces their sum is just all the guys of the form something from you plus something from W okay all sums of little guys that's the sum of the spaces okay and the theorem we had that if we take two subspaces then their sum is also a subspace not just said but subspace can we prove this so here are two more examples two more examples of sums of sums of subspaces so da da so here's example 1 in all of them V is going to be R 3 and all of them and both of them V is going to be R 3 that's that's the ambient space and in example 1 W is going to be I'm going to not write all the details ok in order to save time in space W is going to be all the guys of the form 0 0 C where C is some scalar ok 0 0 C and U is going to be all the guys of the form a be 0 ok so geometrically V is 3 dimensional space u is the xy-plane right all the guys of the form a be 0 they don't have a Z coordinate it's the xy-plane and W is the z-axis all the guys of the form 0 0 C it's just the z-axis do you agree ok what is U Plus w u plus W is all elements of the form something here plus something here so what is it going to be right it's going to be everything it's any element of the form it could have any a any B because it's something here plus something here so it can have any C so this is just do you agree okay so this is their son not their Union there's some okay maybe let's write that this is r3 okay but I want to draw your attention to something that happens here if you take a general element in r3 a general element is a factor of the form ABC right that's a general element in r3 there is only one unique way to write it as an element here there's one unique way to take a general element in R 3 and write it as an element here you have to write it as a B 0 plus 0 0 C there's no other way to write it as an element in this sum do you agree if you take the vector 1 2 5 the only way it can originate as a vector in the sum is 1 2 0 plus 0 0 5 do you see that ok so any vector any vector in r3 can be written as a vector in u plus W in a unique way do you agree here's example two I want to put it on the same board here's example two let's take you is still going to be all the guys of the form a B 0 and W is going to be different now W is going to be all the guys of the form 0 C D ok so when you where allowing anything we want in the first two coordinates and we're forcing zero in the third one in W we want anything we're allowing anything we want in the last two coordinates but we're forcing zero in the first one okay first of all do you agree that both of these are subspaces right geometrically this is the XY plane this is the Y Z plane do you agree if you add two of these you get something and is if you multiply two of these if you multiply one of these by a scalar you against it get something with a zero in the first coordinate there subspaces what is there some what is u plus W in this case okay so in general it's going to be adding somebody like this plus somebody like this right so it's going to be everything of the form a where a can be anything B plus C where B and C can be anything and D in the last coordinate what is this thing it's all of V any element in V can be written like this right for example take C to always be 0 do you agree that that you can you can never get something bigger than the ambient space right you can't suddenly have things that came out of the gear world right so this is just everything this is again r3 do you agree okay but there's something different now it's no longer true that any element in r3 can be written uniquely as something here plus something here so for example if you take the element so now now or not now well it's now but the point is that in this example so there is no longer no longer a unique way to write elements so namely suppose you take let's say let's take a concrete example let's take one two three you can write it for example as 1 2 0 this is an element in you plus 0 0 3 this is an element in w do you agree but you can write this as 1 0 0 plus 0 2 3 do you agree and again this is a legitimate element in you and this is a legitimate element in W so you can write the same element in two different ways and in fact infinitely many ways right you can you can write it as 1 1 0 plus 0 1 3 right and you can write it as 1 negative PI 0 & 0 PI plus 2 3 okay so there are infinitely many ways to write this element as something here plus something here okay so in both cases in both cases the sum of the two subspaces was the entire space in this case the sum was the same in both two examples ok but this one had a property that this one doesn't here we have a unique way of writing something as an element in the sum whereas here we don't do you agree now think for a minute can you see another way of expressing the difference between these two examples what enabled us to write this in two separate distinct ways there was another property of U and W here versus U and W here that allowed us to here in this example to write it in two separate ways and would not allow it here can you see that property right they have an overlap very good and our professional word for an overlap was an intersection here these two spaces don't intersect well right they do intersect but trivially they're only intersection example one is zero do you see that and here their intersection is not trivial right their intersection is everything that the guys that are both in you and in W are all the guys of the form zero something zero do you agree zero something zero is an element here as well as an element here so maybe let's write that let's take a different color I don't want to use another board I want to write it here so in example one let's erase the title giving this an extra line and let's erase this so in this example you intersection W is not empty but it's only the zero vector this is the only vector that's in both of them okay so that's why you don't have any flexibility of moving things from one to the other when you want to write something as the sum of something here plus something there right whereas here the intersection u intersection W is all the guys of the form zero something let's call it a a zero where E is anything right all guys of this form are both in un and W and this is what allows you to move things from one to the other do you agree okay this is what we're going to call a direct sum and this is what we're you're going to call a sum but not a director okay that's the definition that's going to follow now is it clear is the difference between these two examples clear okay so let's do it let's write what we claimed we're gonna write so definition definition u plus W is called a direct sum this is a broader notion from you can even think of it as far up as category theory and I'm not going to even tell you what it is it's a notion that shows up in many many fields of mathematics and here we're only discussing it in the realm of subspaces of vector spaces okay but it's a broad a much broader notion so u plus W is called a direct sum if any element any element in the sum if le if any element in u plus W can be written uniquely you eeeek Li as u plus W where u is in U and W is in W okay so that's the definition example one was a direct sum so example 1 ah was a direct sum example two was not good okay and the serum is the serum is which is which was our observation in in blue in these previous two examples that suppose u and W are subspaces of V U and W and V subspaces then u plus W is a direct sum the sum is always defined and we proved in the previous lesson that it's always a subspace but it's a direct sum namely any element can be written uniquely if and only if remember this notation if and only if u intersection W is just a zero element so this is in fact a criteria you can check this if the only element in the intersection is zero then the sum is a direct sum okay it happened in the examples it doesn't mean that it happens always we need to prove this if we claim that this is a theorem good so let's do that let's prove this so proof we need we need to prove two things right we need to prove two directions we need to prove that if it's a direct sum then the intersection is only zero and then we need to prove separately thief that if the intersection is only zero then it's a direct sum that's what it means to have an if and only if statement that you need to prove this implies this and this implies this it's two statements in one so let's do it so proof I'm going to start by proving this direction so I'm a soon mean that it's a direct sum I want to show that the intersection is zero so take an element any element V in you um intersection W take an element in the intersection what do I want to prove no it's this direction I assume that it's a direct sum so I'm assuming that there's only one way to write every element and I want to prove that the intersection is zero so what I'm going to do is I'm going to take I'm gonna here's the here's V here's you here's W I did the sum it's hard to draw the sum right usually the the way I think of drawing a sum is in fact geometric I take the x axis and the y axis and the sum is the entire plane do you understand what I just did okay I think of a geometrically that's easier to describe rather than with these potatoes that doesn't work here okay so I'm taking a general element in the intersection I want to prove that it's zero necessarily okay so take V and u intersection W let's write it I can always write it as V Plus 0 do you agree that this is the same thing okay do you agree that I can just as well write it as 0 plus V this I can always do okay when you look at it you may be thinking wow these are two different ways to write the same element right here I wrote it as V in U plus zero in W and here I wrote it as zero in U plus V in W right it's clearly an element of the intersection okay but the assumption is that there's only one way to write it as something in U plus something and W that's the assumption do you agree but if there's only one way to write it then the you part in both ways has to be the same thing do you agree so write this and since there is a unique way there is only one way to write V as u plus W where u as in U and W is in W that's the assumption of this direction we're assuming it's a direct sum then the u parts have to be the same thing and the W parts have to be the same thing so we didn't just prove we wanted to prove that V is zero we didn't just prove that V zero we also proved that zero is V do you agree that's of course the same thing well we proved it double okay so V has to be 0 because the u portions of this decompositions has to be the same because there's only one way to write it and in fact 0 has to be T and since there is only as follows that V was nothing other than zero clear what I meant to say about you were supposed to laugh since you didn't I'm assuming that he didn't understand my statement we proved that V is 0 because the u parts are the same but we also proved that 0 is V because the W parts are the same so we proved the claim in double proved it ok it's it's as convincing as can be clear good ok at least you're laughing that's good ok are we done right we have to do the other direction but is this clear okay sometimes it's it's so easy that it gets confusing okay everybody good ok so now let's do the other direction so the other direction maybe let's take a new board so now we're taking two spaces two subspaces unw and we're assuming that they have a trivial intersection there are subspaces so zero is always going to be there but we're assuming that nothing else is there okay so now we're proving this direction so let's write it so we don't get confused assume you intersection W is just the zero vector and let's prove that u plus W is a direct sum okay I forgot to write the notation for a direct sunlight let's go back for a minute to the definition here u plus W is called a direct sum if any element can be written uniquely and now we're proving that this is another way of saying the same thing but there's also notation so let's write it down here you can add it maybe right after the definition add it here notation notation we write u plus W and add a circle around the plus that's the standard notation for a direct sum okay add this please good okay so we're back to the proof of this direction we're assuming now that the intersection is trivial and we want to prove that the sum is in fact the direct sum okay so what do we want to prove we want to prove that any element can be written uniquely that's the definition of a direct sum okay so take an element viii in u+ w that's a general element in the sum and let's show that it can be written in only one unique way as something from here plus something from here okay right suppose you can write it in two different ways write it as u 1 plus W 1 and suppose you can also write it as u 2 plus W 2 two different ways if I prove that u 1 is the same as u 2 and W 1 is the same as W 2 that I'm done that I prove that you tried to write it in two distinct ways but you actually wrote the same thing twice okay do you agree okay so let's show that u 1 so write this and prove and show that u 1 equals u 2 and W 1 equals W - do you agree that this would complete the proof ok so we know that V is U 1 plus W 1 and it's the same as u 2 plus W 2 so this means so this is what we're going to do so this means that I can subtract write subtracting in a vector space just means adding the the inverse write the additive inverse so I can write u 1 minus u 2 equals W 2 minus W 1 do you agree I added you two I added minus u 2 to both sides you to plus minus u 2 is zero disappeared here and u 1 minus u 2 appeared here and then I added minus W 1 to both sides clear so I'm doing this freely as if these we're numbers but they're not they're factors but I'm doing legitimate stuff here clear is it clear why I'm pausing here too our attention to the fact that these are not numbers these are vectors good okay now where is this element u1 minus u2 where does it live it's an e right it's it's two guys from you you and - you right and how do I know that - U is in you how do I know that it's in you how do I know that if I took a vector it's additive inverses there it's a subspace so we only checked that sums are there and okay so for example it's a scalar product of u 2 by minus 1 and it's a subspace therefore it's there good do you understand you have to be careful in these things ok it really seems trivial to say these things if you're thinking in the world of numbers but we're not we're in the world of vector spaces these can be for example two polynomials or two continuous functions or two three by five matrices okay we're in a abstract vector space so is it clear why this isn't you you one isn't you that's how we started you two isn't you that's how we started - you - is in you because it's minus one times you - it's a scalar multiplication and therefore there's some u1 plus - you - isn't you good likewise this is in W do you agree so this dude here which is an element in u equals this dude here which is an element in W so it means that this thing is both in you and in W right but there's only one guy which is both in you and in W which is zero that's the assumption so this whether whether written like this are written like this it's just zero because that's the only guy in the intersection since you intersection w is just zero there's nobody else which is in both clear so u 1 minus u 2 is 0 that implies that U 1 is u 2 and W 2 minus W 1 is 0 that implies that W 1 is W 2 which is what we wanted to prove we took a general element in the sum wrote it in two distinct ways or seemingly distinct ways and ended up with the fact that it was the same clear and this completes the proof okay I want to make a distinction again this is not complicated okay it's not that we somehow you know pulled out of our hats a magic rabbit or or some you know called to our aid some deep theorem from from analysis or anything like that this is everything here is is in an epsilon neighborhood of the definition that that's taken from calculus everything is just using the definitions we didn't do anything complicated it still may be confusing because you have to understand what you're doing you have to understand the concept and what you know what you need to show why everything follows there's some careful understanding of the fact that we're working with vector spaces and everything is well-defined here okay so I'm not saying this is hard I'm just saying read it again make sure that you understand what just happened okay it's not hard it's just it takes understanding to understand it okay and you may need to read it again okay good everybody good okay so you're going to see more examples in in of sums and direct sums and intersections and so on but I'm going to stop our discussion of of sums and direct sums here and we're going to move on to the next topic which we hinted to before suppose we just take a couple of vectors or a few vectors and want to form a subspace that necessarily includes those okay that's what we called linear combinations and that's our next topic

Channel: Technion

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Keywords: Technion, International school of engineering, Dr. Aviv Censor, Algebra 1M

Id: qs240Jhl6Rs

Length: 29min 21sec (1761 seconds)

Published: Mon Nov 23 2015