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Current time:0:00Total duration:15:46

Video transcript

let's say I had the vectors let's say I had the set of vectors and I wonder if that dick let's say one of the vectors is the vector 2 3 and then the other vector is the vector or 6 and I just want to answer the question what is the span of these vectors and let's assume that these are position vectors what are all of the vectors that these two vectors can represent well if you just look at it if you just look and remember the span is just all of the vectors that can be represented by linear combinations of these so it's the set of all the vectors that if I have some constant times 2 times that vector plus some other constant times this vector it's all the possibilities that I can represent when I when I just put a bunch of different real numbers for c1 and c2 now the first thing you might realize is that look this vector - this is just the same thing as 2 times the vector as the two times this vector so I could just rewrite it like this I could just rewrite it as c1 time times the vector 2 3 plus c2 times the vector and here I'm going to write instead of writing the vector 4 6 I'm going to write 2 times the vector 2 3 because this vector is just it's just a multiple of that vector so I could write C 2 times 2 times 2 3 I think you see that these this is equivalent to the 4 6 2 times 2 is 4 2 times 3 is 6 well then we can simplify this a little bit we can rewrite this as just C 1 plus 2 C 2 all of that times 2 3 times our vector 2 3 and this is just some arbitrary constant it's some arbitrary constant plus 2 times some other arbitrary constant so we can just call this C 3 times my vector 2 3 so in this situation even though we started with two vectors and I I said well you know the span of these two vectors is equal to all of the vectors that can be constructed with some linear combination of these any linear combination of these if I just use this substitution right here can be reduced to just a scalar multiple of my first vector and I could have gone the other way around I could have substituted this vector as being 1/2 times this and just made any combination of scalar multiple of the second vector but the fact is is that instead of talking about linear combinations of two vectors I can reduce this to just a scalar combination of one vector and we've seen in our to a scalar combination of one vector especially if they're position vectors for example this vector 2/3 it's 2/3 it looks like this all the scalar combinations of that vector are just going to lie along this line so if I go to 3 it's going to be right there they're all just going to lie along that line right there so find then 4 along this line going in both directions forever and these care if I take a negative values of 2/3 I'm going to go down here if I take positive values and we go here if I get really large positive L is going to go up here but I can just represent the vectors that when you put them in standard form their arrows essentially would trace out this line so you could say that the span the span of my set of vectors let me put it over here the span the span of the vectors let me do it this way the span of the set of vectors 2 3 & 4 6 is just this line here even though we have two vectors they're essentially collinear they're multiples of each other I mean 4 6 is if this is 2 3 4 6 is just this right here 4 6 it's just that longer one right there they're collinear these two things are collinear collinear now and so in this case when we have two collinear vectors in in r2 in r2 they essentially they're span just reduces to that line you can't represent some vector like you can't represent let me do a new color you can't represent this vector right there with some combination of those two vectors there's no way to kind of break out of this line so there's no way that you can represent everything in r2 so the span is just that line there now a related idea to this and notice yet two vectors but it kind of reduced to one vector when you took its linear combinations the related idea here is that we call this set we call it linearly dependent let me write that down linearly dependent this is a linear ly dependent set and linearly dependent just means that one of the vectors in the set can be represented by some combination of the other vectors in the set and a way to think about it is whichever vector you pick that can be represented by the others it's not adding any new directionality or any new information right in this case we already had a vector that went in this direction and when you throw this for six on there you're going in the same direction just scaled up so it's not adding us it's not giving us any new dimension letting us get break out of this line right and you can imagine in 3-space if you have one vector that looks like this and another vector that looks like this two vectors that aren't collinear they're going to define a kind of a two-dimensional space they can define a two-dimensional space let's say that this is the plane defined by those two vectors in order to define r3 a third vector in that set can't be coplanar with those two right if my vector if this third vector is coplanar with these it's not adding any more directionality so this set of three vectors will also be linearly dependent and another way to think about it is that these two purple vectors span this plane span the plane that that they define essentially right anything in this plane going in any direction can be any any vector in this plane when I say span it that means that any vector can be represented by a linear combination of this vector and this vector which means that if this vector is on that plane it can be represented as a linear combination of that vector and that vector so this this green vector I added isn't going to add anything to the span of the of our set of vectors if it and that's because it's this is a linearly dependent set this one can be represented by a sum of that one and that one because this one and this one span this plane in order for the span of these three vectors to kind of get more dimensionality or start representing our three this the third vector will have to break out of that plane it would have to break out of that plane and if a vector is breaking out of that plane that means it's a vector that can't be represented anywhere on that plane so it's outside of the span of those two vectors or it's outside it can't be represented by a linear combination of this one in this one so if you had a vector of this one this one and this one and just those three none of these other things that I drew that would be linearly independent let me let me let me make another let me draw a couple more examples for you that one might have been a little too abstract so for example for example if I had the vectors if I had the vectors 2 3 and I have the vector o-72 and I have the vector and I have the vector 9 5 and I were to ask you are these linearly dependent or independent so at first you say well you know it's not not trivial let's see this isn't a scalar multiple of that that doesn't look like a scalar multiple of either of the other two maybe they're linearly independent but then if you kind of inspect them you kind of see that V if we call this V 1 vector 1 plus vector 2 call this vector 2 is equal to vector 3 so vector 3 is a linear combination of these other two vectors so this is this is a linear linearly dependent dependent set and if we were to show it draw it in in kind of two space and it's just a general idea that or let me see let me draw it in r2 there's a general idea that if you have three two dimensional vectors one of them is going to be redundant as long as well one of them definitely will be redundant for example if we do two three if we do the vector two three that's the first one right there draw it in our standard position and I draw the vector seven to the drop vector seven two so seven two right there I could show you that any point in r2 I could show you that any point in r2 can be represented by some linear combination of these two vectors we could even do a kind of a graphical representation but we I've done that in the previous video so I could write just I could write that the span the span of v1 and v2 is equal to r2 that means that every vector V every every position here can be represented by some linear combination of these two guys now the vector nine five the vector nine five that's what nine five it's right there it is in r2 it is in R to write clearly I just graphed it on this plane it's in our two dimensional real number space or I guess we could call it a space or our in our and our sit in our set r2 it's there it's right there so we just said that anything in r2 can be represented by a linear combination of those two guys so clearly this is an r2 so it can be represented as a linear combination so hopefully you're starting to see the relationship between span and and and linear independence or linear dependence let me do another example let's say I have the vectors let's say I have the vectors let me do a new color it's getting to have the vector and this will be a little bit obvious seven zero so that's my V one and then I have my second vector which is 0-1 that's V two now is this set linearly independent is it linearly independent what can I represent either of these as a combination of the other and really what I see as a combination you would have to scale up one to get the other because there's only two vectors here there's you know if I will trying to add up to this vector the only thing I have to deal with is this one so I can do a scale it well there's nothing I can do no matter what I multiply this vector by you know some constant and you know and add it to itself or scale it up this term right here is always going to be zero it's always going to be zero so nothing I can multiply this by is going to get me to this vector likewise no matter what I multiply this vector by the top term is always going to be zero so there's no way I could get to this vector so both of these vectors there's no way that you can represent one as a combination of the other so these two are linearly independent and you can even see it in if we graph it one is seven zero which is like that let me do it in a non yellow color seven zero and one is 0-1 one is 0-1 and you can I think you can clearly see that if you take a linear combination of any of these two you can represent anything in r2 so the span of these just to kind of get used to our notion of span of v1 and v2 is equal to r2 now this is another interesting point to [ __ ] I said the span of v1 and v2 is r2 now what is the span of what is the span of v1 v2 and v3 in this example up here I already told you already showed you that this third vector can be represented as a linear combination of these two it's actually just these two summed up I can even draw it right here it's just those two vectors summed up so it clearly can be represented as a linear combination of those two so what's its span well the fact that this is redundant means that it doesn't change its span it doesn't change all of the possible linear combinations so it's fan is also going to be r2 it's just that this was more vectors than you needed to span r2 r2 is a two dimensional space and you needed two vectors so this was kind of a more efficient way of providing a basis and I'll be I haven't defined basis formally yet but I just want to use a little I use it a little conversationally and then it'll make sense to you and I define it formally this provides a better basis or this provides a basis kind of a non redundant set of vectors that represent our - while this one right here is redundant so Amit you know it's not a good basis for our - let me give you one more example in three dimensions and then the next video I'm going to make a more formal definition of linear dependence or independence so let's say that I had let's add the vector let me say - zero zero let make a similar argument that I made up there and the vector to 0 0 the vector 0 1 0 and the vector 0 0 7 now are these linked these are an out we're now in r3 right each of these are three dimensional vectors now are these linear dependent or linearly independent so how are they linearly dependent or independent well I can't there's no way with some combination of these two vectors that I can end up with a nonzero term right here to make this third vector right because no matter what I multiply this one by in this one by this last term is going to be 0 so this is kind of adding a new direction to our set of vectors likewise there's nothing I can do there's no combination of this guy in this guy that I can get a nonzero term here and finally no combination of this guy in this guy that I can get a nonzero term here so this set is linearly independent linearly independent and if you were to graph these in three dimensions you would see that they none of these no three these three do not lie in the same plane obviously any two of them line on the same plane but if you were to actually graph it you get to 0 let me say that that's the x-axis that's to 0 0 then you have this 0 1 0 maybe that's the y-axis and then you have 0 0 7 it'll look something like this so it almost looks like your three dimensional axes it almost looks like the vectors ijk they're just scaled up a little bit but you can always correct that by just scaling them down right because we only care about any linear combination of these so these the span of these three vectors right here because they are all adding new directionality is our three is our three anyway I thought I would leave you there in this video I realized I've been making longer and longer videos and I want to get back in the habit of making shorter ones in the next video I'm going to make a more formal definition of linear dependence and we'll do a bunch more examples