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we now have the tools I think to understand the idea of a linear subspace of RN let me write that down then I'll just write it just I'll just always call it a subspace of RN everything we're doing is linear subspace subspace of our n I'm going to make a definition here I'm going to say that a set of vectors V so V is some subset of vectors subset some subset of RN RN so we already said RN when we think about it's really just a really an infinitely large set of vectors where each of those vectors will have n components so you know we I'm going to not formally define it but this is just a set of vectors I mean sometimes we visualize it as multi dimensional space and all that but if we want it to be just as abstract about it as possible it's just all the set it's the set of all of the you know we could call it x1 x2 all the way to xn where each of these where each of the x i's are a member of the real numbers for eyes for all of the eyes right that was our definition of RN it's just a huge set of vectors an infinitely large set of vectors V I'm calling that I'm going to call that a subset of RN which means it's just some you know it's it could be all of these vectors and I'll talk about in a second or it could be some subset of these vectors maybe it's all of them but one particular vector and in order for this V to be a subset to be a subspace so I'm already saying it's a it's a subset of RN so if I were to you know maybe this will help you if I draw all of RN here's big blob so these are all of the vectors that are in RN v is some subset of it it could be all of our and I'll show that in a second but let's just call let's just say this is V V is a subset of vectors is a subset of vectors now in order for V to be a subspace and this is a definition in order if V is a sub V is a subspace or linear subspace of RN of our n this means this is my definition this means three things this means that V contains the zero vector V contains I'll do it really that's the zero vector or you could all move you know this is equal to zero all the way and you have n zeros so V contains the zero vector and this is a big V right there if we have some vector X in V so let me write this so it's X if my vector X is in V is in what is you know if X is one of these vectors that's included in my V then when I multiply X times any member of the reals so if X is in V then if V is a subspace of RN then x times any scalar X times any scalar is also in V this has to be the case and for those of you familiar with the terms this term is called closure if I have any element of a set and then this is closure under multiplication let me write that down under a new color this is closure closure under scalar multiplication scalar multiplication and that's just a fancy way of saying look if I take some member of my set if I take some member of my set and I multiply it by some scalar and I multiply it by some scalar I'm still going to be in my set if I multiplied it by some scalar and I end up outside of my set if I ended up in some with some other vector that's not included in my subset then this wouldn't be a subspace in order for it to be subspace for any if I multiply any vector in my sub in my subset by a real scalar I'm defining the subspace over real numbers if I multiply it by any real number I should also get another member of this subset so this is one of the requirement and then the other requirement is if I take if I take two vectors in my let's say I have vector a it's in here and I have vector B in here so if a so this is my other requirement for V being a subspace if a is in a sorry if a if vector a is in my set V and vector B is in my set V then if V is a subspace of RN that tells me that a a and B must be in V as well so this is closure under addition let me write that down closure closure under addition once again just a very fancy way of saying look if I if you give me two elements that's in my subset and if I add them to each other these could be any two arbitrary elements in my subset and I add them to each other I'm going to get another element in my subset that's what closure under addition means that when you add two vectors in your set you still end up with another vector in your set you don't somehow end up with a vector that's outside of your set so if you if you if I have a subset of RN so some subset of vectors of RN that contains the zero vector contains a zero vector and it's closed under multiplication and addition then I have a subspace so a subspace implies all of these things and all of these things imply a subspace this is the definition of a subspace now this might seem all abstract to you right now so let's do a couple of examples and I don't know if this can examples will make it any more concrete but I think if we do it enough you'll kind of get the intuitive sense of what a space implies I mean it but well let me just do some examples because I want to stay relatively mathematically formal so let's just say I have the the almost trivially basic set let's let's say my set of vectors I only have one vector in it and I have the zero vector so I'll just do a really bold zero there or I could write it like this the only vector in my set is the zero vector and let's say we're talking about r3 so let's say my zero our three looks like that so what I want to know is is my set V is V a subspace subspace of r3 well in order for it to be a subspace three conditions it has to contain the zero vector well the only thing it does contain is the zero vector so it definitely contains the zero vector so the zero vector check now is it closed under multiplication so that means if I take any member of this set there's only one of them and I multiply it by any scalar I should get another member of the set or I should get maybe itself so if let's see there's only one member of the set so the one member of set of the set is the zero vector if I multiply it times any scalar what am I going to get um I get C times C times zero which is zero C times 0 which is 0 and C times zero so it's going to I'm going to get the it's only member but it is closed so it is closed under multiplication under multiplication you can multiply this one vector times any scalar and you're just going to get this vector again so you're going to end up being in your your your zero vector set you're going to sell you that's a check does it closed under addition well clearly if I add any member of this set to each self or I mean there's only one member or to another member of the set there's only one option here if I just add that to that what do I get I just get that I just get it again so it definitely is closed closed under addition addition check so it does turn out that this trivially basic subset of our three that just contains the zero vector it is a subspace it is a subspace maybe a trivially simple subspace but it satisfies our constraints of a subspace you can't do anything with the vectors in it that'll somehow get you out of that subspace or at least if you're dealing with scalar multiplication or addition now let me do one that maybe maybe the idea will be a little clearer if I show you an example of something that is not a subspace let me get my coordinate axes over here so let's say I were to find some subspace let me some subset I don't know whether it's a subspace let me call my set s and it equals it equals all the vectors it equals all the vectors and let me say x1 x2 that are a member of r2 such that I'm going to make a little constraint here such that x1 x1 is greater than or equal to 0 so it contains all of the vectors in R in r2 then that are at least 0 or greater for the first term so if we were to graph that here what do you get we can get anything we can move up or down in any direction right so all we can go up and down in any direction but we're constraining ourselves these are all going to be 0 or greater so all of these first coordinates are going to be 0 or greater so we're going to end this one we can go up and down arbitrarily so we're essentially this subs this subset of r2 r2 is my entire Cartesian plane but this subset of r2 it include the vertical axis often referred to as the y axis it'll include the vertical axis and essentially the first and fourth quadrants if you remember your quadrant labeling so that's the first quadrant and that's the fourth quadrant so my question to you is is s is s a subspace subspace of R 2 so the first question does it contain the zero vector so in the case of r2 does it contain does it contain zero zero well sure it includes zero zeros right there we included we said X is greater than or equal to zero so this could be zero and obviously there's no constraint on this so definitely the zero zero vector is definitely contained in our set s so that that is a check now let's try another one if I add any two vectors in the set is that going to be third is that also going to show up in I set let me just do a couple of examples maybe this isn't to proof let's see if I add that vector to that vector what am I going to get if I put this up here I'm going to get that vector if I add that vector to that vector what am I going to get I could put this one has tail I would get a vector that looks like I would get a vector that looks like that and if I did it any if I did it formally if I add let's say that I have two vectors that are member of our set so let's say the first one is a B and I add it to C D what do I get I get a plus C over this was a D over B plus D so this thing is going to be greater than zero this thing is also going to be greater than zero that was my requirement for being in the set so if both of these are greater than zero and we add them to each other this thing is also going to be greater than zero and we don't care what these these can be anything I didn't put any constraints on the on the second on the second component of my vector so it does seem like it is closed under addition closed under addition now what about scalar multiplication let's take a particular case here let me just well let's just take let's say let's take my a B again I have my vector a B now I can pick any real scalar remember it doesn't you know so any real scalar what if I just multiply it by minus 1 so minus 1 so if I multiply it by minus 1 I get minus a minus B so if I were to draw it visually if this is you know let's say a B was the vector 2 4 so it's like this when I multiplied by minus 1 what do I get I get minus a minus a minus B I get this vector which you can clear visually clearly see falls out of if we view these as kind of position vectors which falls out of our subspace or if you just view it not even visually if you just do it mathematically clearly if this is positive then this is going to end let's say that we assume this is positive and definitely not zero so it's definitely a positive numbers and this is definitely going to be a negative number so when we multiplied it by negative one for really any element of this that doesn't have a zero there you're going to end up with something that falls out of it right this is not a member of the set because to be a member of the set your first component had to be greater than zero this first component is less than zero so this subset that I drew out here the subset of r2 is not a subspace because it's not closed under multiplication or scalar multiplication not closed under scalar multiplication not closed so this is not not a subspace not a subspace of R two now I'll ask you one interesting question what if I asked you just the span of some some set of vectors is that a valid let's say I have the span of some let's say I want to know the span of of yeah no no let's say I've vector v1 v2 and v3 and I'll be you know I'm not even tell you how many elements each of these vectors have is this a valid subspace valid subspace of RN where you know and it's kind of the number of elements that each of these have of RN well let's pick Eadie let's pick one of the elements of let me define let me let me just call let me just call you to be the set or is that the set of all linear combinations is this is the span so let me just define you to be the span so I want to know is you a valid subspace so let's think about it this way let me just pick out a random element of you let me pick out a random element of you actually let me let me zip it does this contain the zero vector well sure if we just multiply all of these times zero that is a so if we just say zero times v1 plus 0 times v2 these are all vectors I didn't write them bold plus 0 times v3 we get the zero vector right we did everything just as zero it out so it definitely contains the zero vector this is a linear combination of those three vectors so it's included in the span now what if I let's just let me just pick some arbitrary member of this span so in order to be a member of this set it just means that you can be represented let me just call the vector X it means that you can be represented as a linear combination of these vectors so you know some combination c1 times v1 plus c2 times v2 plus c3 times v3 right I'm just representing this vector X it's a member of this so it can be represented as a linear combination of those three vectors now is this set closed under multiplication well let's just multiply this time some arbitrary constant what is C times X let me scroll down a little bit what is C times X equal let me do a different constant actually let me multiply it times some arbitrary constant a what is a times X well it's a times c1 times v1 I'm just multiplying both sides of this equation times a times c2 times v2 plus a times C 3v3 right well you could just I mean if this was an arbitrary constant this you could just write this as another arbitrary constant this is another arbitrary constant and this is another arbitrary constant and I want to be clear all I did is I just multiplied both sides of this equation times a scalar but clearly this expression right here I mean I could write this I could rewrite this as you know see 4 times v1 plus C 5 times v2 where this is C 5 this is C 4 plus C 6 times V 3 this is clearly another linear combination of these three vectors so the span is the set of all of the linear combinations of these three vectors so clearly this is one of the linear combinations so it's also included in this and so this is also in you it's also in the span of those three vectors so it is closed closed under multiplication under multiplication now we just have to show that's closed under addition and then we know that the span of we added three here but you can extend it to an arbitrary n number of vectors that the span of any set of vectors is a valid subspace so let me prove that so we already defined one X here let me define another vector that's in U or that's in the span of these vectors and it equals I don't know let's say it equals d1 times v1 plus d2 times v2 plus d3 times v3 now what is what is X plus y if I add these two vectors what are these equal to well I could just add it X plus y means all of this stuff plus all of this stuff and so what is that equal it means if you just add these together you get c1 plus d1 times V v1 plus c2 plus d2 times v2 plus c3 plus d3 times v3 right I just you know you had a v3 here you had a v3 there you just add up their coefficients clearly this is just another linear combination these are just constants again that's an arbitrary constant that's an arbitrary constant that's an arbitrary constant so this thing is just a linear combination of v1 v2 and v3 so it must be by definition in the span of v1 v2 and v3 so we are definitely closed closed under under addition now you might say hey Sal you're saying that the span of any vector is is is is a valid subspace but let me let me show you an example that you know clearly if I just took the span of one vector let me just say let me just define U to be equal to the span of just the vector let me just do a really simple one let's say it's just the vector 1 1 clearly this can't be a valid subspace let's think about this visually so what is a vector 1 1 look like vector 1 1 looks like this right and the span of vector 1 1 is all of this is in its standard position the span of vector 1 1 is all of the linear combinations of this vector well there's nothing else to add it to so it's really just going to be all of the scaled up and scaled down versions of this so if you scale it up you get you get things that look that look more like that if you scale it down you get things that look more like that if you go to the negative domain so with just by multiplying this vector times different values and if you were to grab put them all into a standard position you would essentially get a line that looks like that and you say gee you know that doesn't look like a whole subspace but a couple of things clearly it contains the vector 0 it contains the 0 vector we can just multiply 0 times 1 we can just scale it by 0 right the span is just all of the different scales of this and if there are other vectors you would add it to those as well but this is clearly going to be the 0 vector so it contains the V R 0 vector is it is it closed under multiplication well the span is the set of all of all of the vectors where if you take all of the real numbers for C and you multiply it times 1 1 that is the span so clearly you multiply this times anything it's going to equal another thing that's definitely in your span now the last thing is it closed under addition is it closed under addition so any two vectors in the span could let's say that I have one vector a that's in my span I can represent it as C 1 some scalar times my vector there and then I have another vector B and I could represent it with C 2 times my one vector in my set right there and so what is this going to be equal to this is going to be equal to this is essentially going to be equal to C well for me get a little more space this is going to be equal to C 1 plus C 2 times my vector right this is almost trivially obvious but clearly this is in the span it's just a scaled up version of this this is in the span it's in a scaled up version of this and this is also going to be in the span of this vector because it's just a this is just another scalar we could call that c3 and if you just do it visually you know if I take this vector right there let's say I take that vector and I were to add it to and I were add it to this vector if you put them head to tails you would end up with you would end up with this vector right there in green if you can see it maybe I'll do it in maybe I'll do it in red right there you end up in you will end up with that vector and you could do that any vector plus any other vector on this line is going to equal another vector on this line any vector on this line multiplied by some scalar is just going to be another vector on this line so you're closed under you're closed under multiplication you're closed under addition and you include and you include the zero vector so even this trivially simple span was is a valid subspace and that just backs up the idea that we showed here that in general I mean I could have just made this a set of n vectors right I picked three vectors right here but it could have been end vectors and I could have used the same argument that the span of n vectors is a valid subspace of RN and then I showed it right there