Affine subspaces and transformations - 01 - affine combinations

in these next few videos we'll learn about affine subspaces affine combinations and affine transformations which are very slight generalizations of linear transformations as we'll see so the first definition that we'll need is what an affine combination of vectors is so but to do that will recall what a linear combination is so a linear combination of vectors v1 through VA and RN is a combination of the form lambda 1 v1 so we add up all our vectors with some weights and these weights will take to be real numbers so that's what a linear combination is and closely related to this an affine combination of these same vectors is a linear combination and for short I may often write just using the summation notation oops let's call this not K but J and this goes from J equals 1 to K such that the sum of these coefficients is equal to 1 so it's basically a linear combination but we have an additional constraint on the coefficients so for example when K equals 2 we have two vectors let's say V 1 and V 2 then every such a fine combination is of the form T V 2 plus 1 minus T V 1 where T is a real number and you can look at what this says let's say these two vectors are different let's say V 1 is here and V 2 is here then at T equals zero so this right this is describing the set of all such combinations and when T equals zero this gives me V 1 so at T equals 0 I'm here and when T equals 1 I'm at V 2 and as you vary T over the set of real numbers you get all the points along the straight line through V 1 and V 2 this is very different than the set of all linear combinations of V 1 and V 2 because if let's say the zero vector were here then V 1 would be this corresponding vector V 2 would be this corresponding vector and all linear combinations of these two vectors is actually the plane obtained from V 1 and V 2 that's what the of these two vectors are but all affine combinations is just this line and so just like we can define the span of vectors we can also define the affine span of vectors so the affine span of the vectors v1 through VN we denote it by a FF and it's defined to be the set of all affine combinations so the set of all lambda J VJ such that all of the lambda J's are an R and the sum of them equals 1 so let's look at another example where we take three vectors so let's say V 1 V 2 V 3 and let's just be concrete and let's say we ran R 3 so that we can visualize this a little bit better so there are several cases that we can take just like for linear combinations for instance if one of these vectors was a linear combination of the other then the span of this would be a plane and if all of them are scalar multiples of each other then the span is a line and if they are all the zero vector then we just get the 0 vector and if they're all linearly independent then we get all of our three there are many different cases depending on the relationships between v1 through VN thing happens for a fine span in the sense that it depends on how these vectors are related so let's look at three possible cases so case 1 let's say V 1 V 2 and V 3 are not collinear so this means that all these three points don't lie on the same line so maybe they look something like this like for instance you can take the unit vectors e1 e2 and e3 and r3 and the affine span of these three vectors is equal to the two-dimensional plane containing these vectors and it's not so immediately obvious that that's what happens but let's just think about this if we take v1 and v2 then it includes the affine span of these two vectors which means we have this line through these two vectors is in our a line span and likewise the line through v2 and v3 is here likewise the line v1 through v4 and now that we have all of these lines in here we can also take a line combinations of these points so you can take for instance the affine combination of this point with this point which gives us this line this point with this point which gives us this line and you can see by taking all such combinations all such a fine combinations of these three vectors we can actually get any point in the plane that contains these three points in case two let's imagine that v1 v2 v3 are collinear but at least two our distinct so in this case so I'm assuming that at least two so either the possibilities are something like they're all different but they lie on the same line in which case the affine span of these three points is equal to the straight line through those two points those three points or the other case is the affine span if two of them happen to coincide and we just have two points but I'm assuming that they're collinear and at least two are distinct so we also get the straight line through those two points and the final case case three is when all those vectors are exactly the same vector and when this happens we only have a single point and all affine combinations of a single point is just that point itself so these are some of the basic constructions that you can do with vectors besides just taking linear combinations you can also take a fine combinations there's yet another type which we won't discuss is if you require that the sum of these coefficients adds up to one but they're also not just real numbers but they're strictly non-negative so they have to be at least zero and that's called a convex combination which is a closer related idea and in the case of these three vectors for instance it would be the triangle whose three vertices are those three vectors that we had here and in this case if we took convex combinations it would be the interval between these two farthest end points and in this case we would have the same situation as we had here where we would just get a single point

Channel: Arthur Parzygnat

Views: 12,529

Rating: 4.9783196 out of 5

Keywords: video lectures, math, physics, category theory, linear algebra, least squares, error-correcting codes, error-correction, linear regression, support vector machines, machine learning, convex, affine transformation, affine, combination, computer science, geometric, composition

Id: fWRm9dISpNk

Length: 9min 46sec (586 seconds)

Published: Tue Dec 24 2019