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# Linear combinations and span

## Video transcript

one term you're going to hear a lot of in these videos and in linear algebra in general is the idea of a linear combination linear combination and all a linear combination of vectors are oh they're just a linear combination I mean let me show you what that means so let's say I have a couple of vectors v1 v2 and it goes all the way to VN and there are Lynn you know can be an r2 or RN let's say that they're all they're all in RN you know they're in some dimension of real space I guess we could call it but the idea is fairly simple a linear combination of these vectors means you just add up the vectors it's some combination of the sum of the vectors so v1 plus v2 plus all the way to VN but you scale them by arbitrary constants so you scale them by c1 c2 all the way to CN where everything from c1 to CN are all a member of the real numbers that's all a linear combination is let me show you a concrete example of linear combinations let me make the vector let me define the vector a to be equal to and these are all bolded these purple these are all bolded just so those are vectors but sometimes it's kind of owners to keep bolding things so let's just said to find the vector a to be equal to 1/2 and I define the vector V to be equal to 0-3 what is a linear combination of a and B well I could just I just gives you any constant times a time plus any constant times B so it could be it could be zero times a plus what could be zero times a plus zero times B which of course would be what that would be zero times a would be zero zero and zero zero that be the zero vector but this is a completely valid linear combination and we can denote the zero vector by just a big bold zero like that we could also this is almost I could do three times a I'm just picking these random these numbers at random three times a plus we do a negative number just for for fun so let's do plus minus two times B what is that equal to let's let's figure it out it's it's let me write it out it's minus is three minus two times zero so minus zero and it's 3 times 2 is 6 6 minus 2 times 3 so minus 6 so it's the vector it's the vector 3 0 this is a linear combination of a and B I could keep putting in a bunch of random real numbers here and here and I'll just get a bunch of different linear combinations of my vectors a and B if I had a third vector here if I its vector C and maybe that was just you know 7/2 then I could add that to the mix and I could you know I could throw in plus 8 times vector C these are all just linear combinations now why do we why do we you know why do we just call them combinations why do we have to add that little linear prefix there because we're just scaling them up we're not multiplying the vectors times each other we're not we haven't even defined what it means to multiply a vector and there's actually several ways to do it but you know we can't we can't square a vector and we don't we haven't even defined what this means yet but this this would all sudden make it nonlinear in some form so all we're doing is we're adding the vectors and we're just scaling them up by some scaling factor so that's why it's called a linear linear combination now you might say hey Sal why are you even introducing this idea of a limit linear combination because I want to I want to introduce the idea and this is an idea that that confounds most students when it's first taught I think it's just a very nature that it's taught over here I just kept no I don't want to do that I just kept putting different numbers for the weights I guess we could call them for you know C 1 and C 2 of this combination of a and B right let's ignore C for a little bit I just put in a bunch of different numbers there but it begs the question what is the set what is the set of all of the vectors I could have created and this is just one member of that set but what is the set of all of the vectors I could have created by taking linear combinations of a and B so let me draw a and B here maybe we could think about it visually and then maybe we could think about it mathematically so let's say a and B so a is 1/2 so 1/2 looks like that that's vector a let me do a vector B in a different column do it in yellow vector B is 0-3 so vector B looks like that 0 3 so what's the set of all of the vectors that I can represent by adding and subtracting these vectors and we saw we could still if you multiply both by 0 and add them to each other we end up there if we take 2 times 3 times a that's a the equivalent of scaling up a by 3 so you go 1 a 2 a 3 a so that's 3/8 3 times a will look like that 3 so this vector is 3 a and then we add it to that to be all right oh no we subtracted to be from that so minus B looks like this minus B looks like this minus 2 B minus 2 B looks like this this is minus 2 be all the way in standard form standard position minus 2 B so if you add 3 a to minus 2 B we get to this vector 3 a to minus 2 B you get this vector right here and that's exactly what we did when we've solved it mathematically you get the vector 3 0 you get this vector right here 3 3 0 but this was just one combination one linear combination of a and B I could have instead of multiplying a times 3 I could have done I could have multiplied a times 1 and a half and just gotten right here so one and a half a one and a half a minus 2 B minus 2 B would still look the same it would look like something like this it would look something like it would look let me make sure I'm doing that it would look something like this and so our new vector that we would find would be something like this so I just showed you I could find this vector with a linear combination I could find this vector with a linear combination and actually turns out that you can represent any vector in R to any vector in r2 with some linear combination of these vectors right here a and B and let's just just let's just let's just think of an example or maybe just try to a mental visual example wherever we want to go wherever we want to go that we could go arbitrarily we could scale a up by some arbitrary value so this is some scaled up so this is some weight on a and then we can add up arbitrary multiples of B B goes straight up and down so we could add up operatory multiples of B to that so we could get any point on this line right there now if we scaled a up a little bit more if we scaled up a little bit more and then added any multiple B we get anything on that line if we multiply a times a negative number and then add it a B and either direction we get anything on that line we can keep doing that and there's no reason why we can't pick an arbitrary a that can fill in any of these gaps if this is if we want to point here we just take a little smaller a and then we can add all the B's that fill up all of that line so we can fill up any point in r2 with the combinations of a and B so what we can write is what we can write here is the span let me write this word down the span the span of the vectors a and B so let me write that down of the vectors a and B it equals r2 or equals all the vectors in r2 which is you know it's all the tuples r2 is all the tuples made of two ordered tuples of two real numbers so it equals all of our two this just means that I can represent any vector in r2 with some linear combination of a and B and you're like hey can't I do that with any two vectors well what if a and B what if a and B were the vector let's say the vector two two was a so a is equal to two two and let's set say that B is the vector let's say the B is vector minus 2 minus 2 so B is that vector so B is the vector minus 2 minus 2 now can I represent any vector with these well I can definitely Rep I can scale a up and down so I can scale a up and down to get anywhere on this line and then I can add B or anywhere to it and I could and B is essentially going in the same direction it's just in the opposite direction but I can multiply by negative and go anywhere in line so any combination of a and B will just end up on this line right here if I draw it in standard form it'll be it'll be a vector with the same slope as either A or B or same inclination whatever you want to call it I could never there's no combination of a and B that I could represent this vector I could represent vector C I just can't do it I can add in standard form I could just if I just I could just keep adding scale up a scale up B put them heads to tails I'll just get to stuff on this line I'll never get to this so in this case the span and I want to be clear this is for this particular a and B not for the a and B ID for this blue a and this yellow B the span here is just this line it's just this line it's not all of our two so this isn't just some kind of you know statement you might have when I first did it with that example it's like okay any two vectors can I represent can't can any two vectors represent anything in r2 well no I just showed you two vectors can't represent that and what if I mean you know what is the span of what is the span of the zero vector I'll put even a cap over the zero vector I'll make it really bold well the zero vector is just zero zero so I don't care what multiple I put on it I could put the span of it is all of the linear combinations of this so essentially I could put arbitrary real numbers here but I'm just going to end up with the zero zero vector so the span of the zero vector is just the zero vector the only vector I can get with a linear combination of this the zero vector by itself is just the zero vector itself likewise if I take the span of just you know let's say I go to this let's say I go back to this example right here are my a vector was right like that let me draw it in a better color my a vector look like that if I were to ask just what the span of a is it's all the vectors you can get by creating a linear combination of just a so it's really just scaling you can't even talk about combinations really so it's just C times a all of those vectors and we saw on that in that in the video where I pair where I parametric eyes door to show the parameter parametric representation of a line that this the span of just this vector a is all of the line is the line that's formed when you just scale a up and down so span of a is just a line you have to have two vectors and they can't be collinear in order to span all of our two and I haven't proven that to you yet but we saw with this example if you pick this a and this B you can represent all of our two with just these two vectors now the two vectors that you're most familiar with two that span r2 or if you take a little physics class you have your I and J unit vectors I and J I and J and in our notation I the unit vector I that you learn in physics class would be the vector 1 0 so this is I that's the vector I and then the vector J is the unit vector 0 1 this is what you learned in physics class let me do a different color this is j j is that and you learned that they're orthogonal and we're going to talk a lot more about what orthogonal t means but in our in our traditional sense that we learn in high school it means that they're 90 degrees but you can clearly represent any angle or any any vector in r2 by these two vectors and the fact that they're orthogonal makes them extra nice and that's why these form and i'm going to throw out a word here that i haven't defined yet these forms the basis these form a basis for r2 and that in that you can represent anything in r2 by these two vectors long I'm not going to even define what basis is that's going to be a future video but let me make let me just write the formal math you definition of span just so you're satisfied so if I were to write the span the span of a set of vectors V 1 v2 all the way to VN that just means the set the set of all of the vectors where I have c1 times v1 plus c2 times v2 all the way to CN let me scroll over all the way to CN VN so this is a set of vectors because there I can pick my C eyes to be any member of the real numbers and where's you know and that's true for i from so I should write for I being anywhere between 1 and n all I'm saying is is that look each I can multiply each of these vectors by any value any arbitrary value real value and then I can add them up and now the set of all of the combinations scaled up combinations I can get that's the span that's the span of these vectors you can kind of view it as the space of all of the vectors that can be represented by a combination of these vectors right there and so the word span I think it does have an intuitive sense I mean it all if I if I say that you know my first example I sold those two vectors span or a and B spans are - I wrote it right here that tells me that any vector in r2 can be represented can be represented by a linear combination of a and B and actually just in case that visual kind of pseudo proof doesn't do you justice let me prove it to you algebraically I'm telling you that I can take let's say I want to represent you know I have some let me rewrite my A's and B's again so this was my vector a it was one two and B was 0 3 let me remember that so my vector a a is 1 2 and my vector B was 0 3 now my claim was that I can represent any point let's say I want to represent some arbitrary point X in R 2 so its coordinates are X 1 and X 2 I need to be able to prove to you that I can get to any x1 and any x2 with some combination of these guys so let's say that my combination I say I say c1 times a plus c2 times B has to be equal to my vector X so let me show you that I can always find a c1 or c2 given that you give me some X's so let's just write this right here with the actual vectors being represented in their kind of column form so we have c1 times this vector plus c2 times the B vector zero three should be able to be equal my X vector should be able to be equal to my x1 and x2 where these are just arbitrary so let's see if I can set that to be true so if this is true then the following must be true c1 c1 times 1 plus 0 times c2 must be equal to x1 we just get that from our definition of multiplying vectors time scalars and adding vectors and then we also know that 2 times c2 2 times c2 or C sorry 2 times C 1 times 2 C 1 times 2 plus C 2 times 3 3 C 2 should be equal to x2 now if I can show you that I can always find C 1 s and C 2 is given any X's and any X 1 and X 2 s then I've proven that I can get to any point in r2 using just these two vectors so let me see if I can do that so this is just a 2 to unknown a system of two unknowns this is just 0 we can ignore it so let's multiply this equation up here by -2 and put it two here so we get minus 2 C 1 I'm just multiplying this times -2 we get a 0 here plus 0 is equal to minus 2 X 1 and then you add these 2 you get 3 C 2 right these cancel out you three let me write in a different color you get 3 C 2 is equal to X 2 minus 2 X 1 or divide both sides by 3 you get C 2 is equal to 1/3 X 2 minus X 1 now we'd have to go substitute back in for for C 1 well we have this first equation right here that's C 1 this first equation is it says C 1 plus 0 is equal to X 1 so C 1 is equal to X 1 so that one just gets us there so C 1 is equal to X 1 so you give me any point in R 2 these are just two real numbers and I can just perform this operation and I'll tell you what waits to apply to a and B to get to that point if you say ok what combination of a and B can get me to the point let's say I want to get to the point let me go back up here so well it's way up there let's say I'm looking to get to the point 2 2 so X 1 is 2 let me write it down here say I'm trying to get to the point the vector 2 2 what combinations of a and B can be there well I know that C 1 is equal to X 1 so that's equal to 2 and C 2 is equal to 1/3 times 2 minus 2 so C so 2 2 minus 2 is 0 so C 2 is equal to 0 so if I want to just get to the point 2 2 I just multiply oh I just realized this was looking suspicious I made a slight error here and this was good that I actually tried it out with real numbers over here when I had 3 C 2 is equal to X 2 minus 2 X 1 I got rid of this 2 of over here there's a 2 over here I divide both sides by 3 I get 1/3 times X 2 minus 2 X 1 and that's why I was like why this doesn't list this this is looking strange not to take a moment of pause so let's go to my my my corrected definition of C 2 C 2 is equal to 1/3 times X 2 so 2 minus 2 times X 1 so minus 2 times 2 so it's equal to 1/3 times 2 minus 4 2 minus 4 which is equal to minus 2 s is equal to minus 2/3 so if I multiply so if I multiply 2 2 times my vector a minus 2/3 times my vector B I will get to the vector 2 2 and you can verify it to your for yourself 2 times 2 times my vector a 1 2 minus 2/3 times my vector B 0 3 should equal to 2