24 - Intersections and sums of subspaces

oh there we are intersections in sums of subspaces so let's start with intersection suppose we have a big space V that's all the board okay I'm just gonna make a an abstract drawing to kind of demonstrate the ideas and here's a subspace of V let's call it u okay maybe let's draw V so here is V V is a big vector space and U is a subspace okay and suppose we have another subspace let's call it W here's W okay so this is in general just abstract drawing okay so what is the intersection of well I made it kind of small let's make it a bit bigger so we have some room to navigate in so here's W okay so the intersection is just this part right this is this is U intersection W all the guys that are both in you and in W do you agree okay is you intersection W a subspace okay you as a subspace W is a subspace is the intersection of subspace okay so don't guess we need to check we need to check if the three conditions for being a subspace are satisfied okay so the theorem is yes they are okay theorem if u and W are subspaces subspaces of V then so is you intersection okay so let's check that how do we prove such a theorem let's look at the previous board which I left from the previous clip in order to determine if something is a subset we need to verify that it's non-empty namely check that zero is there that if you take two guys their sum is there and if you take any element its any skill or multiple if it is there right these are the three things that you have to check so let's verify those so the first thing we need to check is that zero is in the intersection well zero has to be in u because U is a subspace zero has to be in W because W is a subspace so if it's both in U and and W it has to be somewhere here in the intersection okay so one how do we write something like this that this is always kind of a tricky part maybe the idea of what you need to write is very clear but how do you write it okay so here's how we write it zero belongs to you and zero belongs to W because both are subspaces then they have to contain zero therefore zero belongs to you intersection now believe good and there's nothing more to write here okay well if you want to be completely formal you have to write what I set out loud since U and W are subspaces okay okay what's the second thing suppose we take any element here any element here let's call it I don't know let's call it little V okay suppose now two was the closure under addition sorry so suppose we take v1 and v2 in the intersection okay we took two guys here is there some necessarily here maybe we took an element here and another element here but their sum is out here okay so we need to check that v1 plus v2 is also in the intersection but what does it mean that v1 and v2 are in the intersection right it means that they're both in you and in W okay so if they're in the intersection in particular v1 and v2 belong to you for example because they're in the intersection right but if they're in you and you as a subspace their sum is also in you right good so now we know that we took two guys here we took two guys here here's uh let's say Z 1 and here's V 2 now we know that their sum is somewhere inside you okay but for the exact same reason their sum is also going to be inside W because W is a subspace as well okay so V 1 and V 2 belong to W this follows from this as well because they're in the intersection and therefore their son belongs to W now we know that their sum is also here and if it's both here and here it's in the intersection so from these two properties V 1 plus V 2 belongs to you intersection W good this U is ugly good clear and what's the last thing that we need to check right so suppose we take any element in the intersection V and alpha any scalar in the field F we're working over a general field then alpha times V belongs to you because V is in U and alpha times V also belongs to W because V is in W and both of these are subspaces therefore alpha V belongs to the intersection good so all three criteria for being a subspace are satisfied so the intersection is always a subspace okay what about the union okay so we have two subspaces U and W of a bigger space V what about the Union the union is everything that's either in you or in W this is the union right you know at a union of two senses is it a subspace hint it's not in the title so the answer is probably no why not what would feel so let's just start thinking naively if it is zero gonna be here wait wait wait wait slowly is zero gonna be there yeah because zero has to be in both so zero is even in the intersection and the intersection is clearly included in the Union right so zero is there what about scale of pluff I'm leaving the the tricky property for Less but what about scalar multiplication okay if you take a guy in the Union it came from you or from W right so if you multiply it by a scalar if it was originally in you multiplied by a scalar still in you it was originally in W multiplied by a scalar it's still in W so it's still going to be in the Union so this property is gonna hold and the property that's gonna fill is the addition if you take a guy from year and a guy from here and add them there's no good reason for you to still be in the Union okay if you take two guys from you and add them they're gonna be inside because you use a subspace if you take two guys from W and add them they're gonna be inside but if you take somebody from here and somebody from here there's no axiom that tells you that their sum is not going to be out here okay now that might sound convincing there's no axiom that tells you but it's not as convincing as we want in mathematics if you want to say that a theorem is not true what do we need in order for it to be right we need a counter example then nobody can argue right so let's give a counter example to the fact that a union is not necessarily a subspace so remark u Union W is usually it's usually with double L or one L double elf I think I made a mistake previously u Union W is usually not a subspace so here's a counter example let's take V so this is a counter example let's take V to be R - okay the if you want to think of it geometrically it's the plane okay let's take u you this is going to be one subspace it's gonna be all the guys of the form a 0 for some 8 so you should convince yourself for a minute that this is indeed a subspace any multiple of a vector of the form a 0 is going to be of the form a 0 add two guys like this it's going to be of the form still going to have 0 into second coordinate and zero is there zero zero right so this is a subspace what what does it represent geometrically what is this subspace right this is the x-axis do you see that so this is the plane the XY plane if you want this is the x-axis and let's stay W to be all the guys of the form 0 B where B is any scalar and this is the y-axis right this is if we think geometrically do you agree what is the union of these guys the union not the intersection the intersection is only the zero vector zero zero and that is a subspace it's that what we call the trivial subspace okay but what is the union of these guys that's not true the union is only the union is only elements which are here or here okay you can throw in all these guys and you throw in all these guys you're never gonna get an element of the form a B where both a and B are not zero weight that sums that's something else but the Union so the Union so let's maybe write what you commented before W intersection u is just the zero vector and this is a subspace that's the previous example the intersection this is a theorem right and it of course holds here as well but W Union U is geometrically it's just the X YZ it's just the axes it's this big plus not the entire plane and not I don't know how to and not this just the exes themselves okay so for example let's continue here so for example 1 0 belongs to the Union right because this is an element of U right and 0 1 belongs to the Union because it's an element of W but there's some is the vector 1 1 and this does not belong to the Union because it's not an element of you and it's not an element of W thank you does not belong to the Union ok so the union is not closed under addition clear ok so note that what fails note that closure under addition is what fails right this is the general phenomena and then the seeds in this example you see it concretely ok so if you want to take two spaces and somehow form a bigger space that includes the two okay what we need is to throw in all the sums of all the vectors in both of them that's not the Union the Union doesn't contain sums it just contains both ok and this is gonna be a notion called the sum of two vector spaces okay so that's the other thing in the title what is a sum effector space is and that's exactly what we're gonna need in order to make a bigger subspace containing two given subspaces there's some so-so definition if u and W are subsets of V when I write this all I mean is that there are subsets no no claim about being subspaces okay just subsets u plus W this is a sum of two sets okay is by definition the set whose elements are all sons of elements from these two sets so there are all the guys of the form u plus W we're using U and W is in W okay so this is the sum of two sets of two subsets okay so this this assumes that V has some additive structure on it otherwise what is u plus W okay so if he has some let's say that V is a vector space all we need for this is for V to being for example a group right a group has an addition property but let's say that V is a vector space so this is what we call the son u plus W is called the sum of U and W okay and by the way do you see that the sum the descent contains you as a subset right if these are if these are indeed subspaces then W contains zero and then we take all the guys of the form u plus zero right we get u itself and if we take all the guys of the form 0 plus W we get W itself so it does contain U and W but it contains more more than just the Union it contains all sums of these guys okay so the theorem is U plus W or you have to say if u and W are subspaces of V then so is U plus ok and now we need to prove this so how do we prove that something is a subspace we're kind of already accustomed to this routine we need to check that it's not empty namely check that 0 is there check that if you add two guys they're gonna be there and check that if you multiply you somebody by a scalar it's gonna be there okay so what's the easiest thing in this case well in fact all three are completely trivial let's just write it I did it was set up so that addition holds right do you agree and scalar multiplication is easy just as well and zero is easy so everything's easy sometimes things are easy not they're not always hard sometimes they're hard but sometimes they're easy so proof of this theorem so the first thing zero we can write it as zero plus zero this is a very deep fact right but this is zero it belongs to you this is zero it belongs to W because U and W are subspaces so this is an element of U plus W so zero is an element of u plus W do you agree ok now we need to check scalar multiplication so let's take a guy in u plus W it looks like that that's a general element in u plus W let's multiply it by some alpha is this now in u plus W write this equals alpha u plus alpha W how do we know that we have this distributive law right it's in V everything happens inside V and V is a vector space right so we know that this holds this is now a guy in you this is a guy in W because there are subspaces so this belongs to you plus W clear and finally suppose we take two guys let's say u 1 plus W 1 here's an element of u plus W take another guy you 2 plus W 2 this is another element in U plus W is there some in u plus W well we can rewrite this as u 1 plus u 2 plus W 1 plus W 2 this is the associative property of addition in the ambient space V right and now we wrote it as an element in you this belongs to you and this belongs to W and therefore their sum is an element in u plus W and that's it this completes the proof we forgot to write squares how much is previously good clear okay so and I'm saying it's easy it's easy if you the only thing you really have to understand in order to write this proof is what we need to write if you understand what we need to write then actually writing it is easy that the only concept you have to grasp here is what do we need to show what does it mean for this to be a subspace and the theorem that tells you these are the only three things that you have to check okay let's do some examples some examples so example one is going to be the the previous example that we had remember we had it's still on the board look at this at this board a second you was V the ambient space was are two you as the x-axis all the guys of the form a 0 and W was the y-axis all the guys have deforms zero B right what is their son what is the sum of U and W it's V right it's all the guys of the form something in U plus something in W so it's any vector a B any vector a B is something in you plus something in W so there's some is the entire r2 in this case okay so let me go back here in the previous example in the previous example u plus W is the set of all elements of the form something in U plus something in W right that's the definition of a sum of two vector spaces something in you this is a general something in u plus something in W okay but these are just these sums are just any vector of the form a B so this is in fact V itself which was or two and clearly it's a subspace it's the entire space good clear know what what are elements of what our sums of these two guys you can add if you want to be completely precise and clear and show all your work you would maybe write this and then you can say well this is precisely our to which was good okay here's another example now we can erase this example here's another example let suppose V is r3 now so that you can think of it geometrically as this three dimensional space okay 3d space and let's say that U is all the guys of the form a 0 0 where a is a scalar and W is all the guys of the form 0 B 0 where B is a scalar so if you think of this geometrically U is filled the x-axis and V and W is still the y-axis and what is U plus W now it's all the guys of the form something here plus something here so it's all the guys of the form a B 0 do you agree where a B are any scalars this is not our to remember we had this discussion this is not r2 because it's still three tuples okay but it's sitting inside our three as the plane geometrically do you agree okay and this is of course a subspace okay good okay yeah so let me think if I want to break this into two yeah okay so I want to do two more examples but I'm gonna do them in the next video because they're gonna lead to one further notion which is the notion of a direct sum okay so it's a refinement of this notion of being the sum it's called a direct sum and the next two examples are gonna lead to it so I'm gonna do them in a minute after we press stop and press play again okay questions about this is the notion of of sum of two spaces clear and the difference between just the union of both spaces okay this is clearly a much bigger set than just a Union okay the Union is only guys of the form a 0 0 and guys of the form 0 B 0 but nothing of the form a B 0 is going to be in the Union they the sum is exactly going to have all these a be zeroes ok good ok so let's stop this one here

Channel: Technion

Views: 43,553

Rating: 4.9414892 out of 5

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

Id: Rcj1-E3SAhs

Length: 27min 5sec (1625 seconds)

Published: Mon Nov 23 2015