Gram-Schmidt Orthogonalization

I want to talk to you about how to find an orthonormal basis for a vector space or for a subspace of a vector space for example this purple plane here that's a subspace of r3 and it is going through the origin so that does indeed make it a subspace of r3 it's not just any other plane and maybe my goal is to find an orthonormal basis for it it certainly has a basis of two vectors but I want an orthonormal basis so the reason I might want to do that one behind the axes here I might have a vector here this white vector and I want to project it onto that purple plane and there's a there's a pretty straightforward way of doing that if indeed I have an orthonormal basis for that purple plane it just involves doing a couple projections on to the basis vectors and that of course looks like this the projection vector would be this and it's perpendicular component would be this and the point is how do I get these two blue vectors from that white one and again if I have an orthonormal basis for the purple plane there's a way of doing that so but the problem is that it might not be totally obvious how to get the North a normal basis but it might be rather straightforward to get any old basis for this purple plane I might have already been given that maybe that's how the plane was described or maybe if I had an equation for the plane I could do a little arithmetic and try to find a couple of vectors which are a spanning set for that plane but you can see if I come up with say two basis vectors say these two green ones they're not orthogonal to each other and they're probably not even of length one so they're not an orthonormal basis but they are a basis and the question is can I work with that somehow can I kind of coax those into being an orthonormal basis by altering them in some way so there's a way of doing that but let me show you how that works in two dimensions here it's easier to see what's happening if we do it in 2d so let's move here to 2d I have two vectors here V 1 and V 2 those are indeed a basis for R 2 of course we already have an obvious orthonormal basis for R - that's II 1 and E 2 1 0 and 0 1 but the point is I want to think about that more general situation where have a basis already I want to try to tweak it somehow and kind of convert it or get the north or normal basis from that so let's start with having v1 and v2 here and see if I can kind of alter them in some way to get an orthonormal basis so v1 and v2 are the vectors as shown those are not of length 1 v1 has length what is that square root of 10 I think and and they're not even orthogonal to each other either so here's the process now let me get rid of the labels here I don't really want to think about the numbers so much I want just think about the geometry of this and let's go through the idea you what we're gonna do is we're gonna keep v1 this vector v1 we're gonna we're gonna hang on to that one we're gonna keep that in our basis except it's not of length 1 so I have to fix that its length is square root of 10 right now or something like that so what I'm gonna do is I'm going to shorten it I'm going to shorten it down to a unit vector and you would say you're normalizing the vector when you do that so you know how to do that you multiply by 1 over the magnitude and that would convert it to this orange vector here so then I'll let me get rid of v1 so I'm replacing it with u1 so you want us just a shortened version of v1 and then what we're gonna do is we're going to take v2 and we're gonna extract out of that something which is orthogonal to u 1 and that'll be our other basis vector what's kind of important right now about v2 is that it's not parallel to u1 it couldn't be because v1 and v2 our basis for r2 so v2 is definitely not parallel to you want that means that v2 has a parallel component to you want and a perpendicular component of v1 excuse me to u1 which is nonzero so you know how to do this I'm going to connect v2 down onto the U 1 vector I guess what I really mean when I say that when we say we're projecting a vector onto anothers we're projecting the blue vector down onto the space spanned by u1 this line here so that would look something like this we drop a perpendicular down this is geometrically what's happening and then the projection vector looks like this this green vector here's the action of the v-2 onto the line spanned by u1 and then the perpendicular component of that is obtained by a subtraction right we're just gonna take v2 and subtract off its perpendicular component and then we'd have v2 perpendicular there no no if I said that right we're going to come subtract the parallel component from v2 to get the perpendicular component of v2 and that said this is going to be our vector that we're going to use for the basis feed to perpendicular let me draw that though over coming out of the origin because that might be a better place to see it and hide some of these other components here and u 1 and v2 perpendicular are now indeed orthogonal to each other and they serve as a basis for r2 there's only one last thing to do v2 perpendicular is not of length 1 and it's an easy conversion to normalize that you divide by its length and you get a vector which is indeed of length 1 and that would give us a vector we can call you two so now u 1 and u 2 are both of length 1 and they're perpendicular to each other so there are an orthonormal basis for r2 and again remember how we did that we started with two vectors v1 and v2 from behind this L here we started with v1 and v2 and we kind of coax them into giving us an orthonormal basis for r2 we kept v1 we just shortened it down and then we took the part of v2 which was perpendicular to that first u1 and that's called the gram-schmidt orthogonalization process for getting an orthonormal basis starting from some other basis that you have so now remember that these vectors were very specific v1 v2 had specific components to them maybe you can kind of go back watch the video again and see if you can actually work out the numbers here what exactly are you one in u2 there as vectors follow through with the projections and subtraction and all that and see what you get

Channel: Dan Gries

Views: 12,761

Rating: 4.9669423 out of 5

Keywords: linear algebra, gram-schmidt, orthogonal

Id: KOkuTXrv5Gg

Length: 6min 37sec (397 seconds)

Published: Fri Nov 10 2017