Coordinates with respect to a basis

let's say I've got some subspace of RN let's say V is a subspace is a subspace of R N and let's say the set B do in blue let's say the set B is a basis for V so it's got a bunch of vectors in it let's say it's got v1 v2 all the way to VK so you can see we have K vectors so V is a k-dimensional is a K dimensional subspace so that means that means that if I have some vector a let's say I have some vector a that is a member of my subspace that means that I can represent a is a linear combination of these characters right there so I could write a as being equal to some constant times my first basis vector plus some other constant times my second basis vector and then I could keep going all the way to a case constant times my Cate basis vector now I've used the term coordinates fairly loosely in the past and now we're going to have a more precise definition I am going to call I am going to call these these constants here I am going to call these constants here so I'm going to call c1 c2 all the way through CK I'm going to call them the others a new color the coordinates coordinates coordinates of a with respect with respect to our basis B with respect to B and we could also write it like this we could also write we could also write I have my vector a so I have my vector a but if I wanted to write my vector a with in coordinates with respect to this basis set B I would write it like this put brackets around it and put the basis set right there and this says I'm going to write this in coordinates with respect to this Bay set so that I will then write it like this I will then write it like this I'm just going to put these weights there these constant terms on the on the linear combination that I have to get of my basis vectors to get a so C 1 C 2 all the way to C K now there's one slightly interesting thing or maybe very interesting thing to point out here V is a basis of RN so anything in V is also going to be in RN but V has K vectors has dimension K and that K could be as high as n but it might be something smaller maybe this is a you know maybe we have two vectors in r3 in which case B would be a plane in r3 but we can abstract that further dimensions but when you specify something that is in our subspace with respect to its basis notice you only have to have that many the dimension of your of your subspaces you only have to have that many coordinates so even though a is a member of RN I only have to give it K coordinates because essentially you're giving it positions within within let's say if this was a plane within the plane that is your subspace let me make this a little bit a little bit more concrete let me do some examples so let's say we have some subspace let me clear this out let's say I have a couple of vectors let's say V 1 is the vector 2 1 and let's say V 2 is the vector is the vector 1 2 now you might immediately see that the basis or the set of V 1 and V 2 this is a basis for R 2 this is a basis for R 2 which means that any any vector in R 2 can be represented as a linear combination of these guys I could do a visual argument or we also know that look R 2 is 2 dimensional and we have two basis vectors right here and they are linearly independent you can verify that in fact the easiest way to verify that is if you just take - one and one - and you put it in reduced row-echelon form you're going to get the at two by two identity matrix you're going to get one zero zero one and that lets you know that this guy and this guy are both basis vectors so that's all review we've seen that before but let's visualize these let's visualize these guys so if I were to just graph it than the way we normally graph these vectors what does two one look like let me draw some axes here let me draw it let me do it in a different color let me let's say that is my vertical axis and this is my horizontal axis and two one might look like this so we're going to go out one two and then we're going to go up one so that is our vector one right there that is two one that's our vector 1 and then 1 2 1 2 might look like or it does look like this if I draw in standard position 1 let me go up to 1/2 looks like this 1 2 looks like this so when we talk about coordinates with respect to this basis let me pick some other let me pick some member of r2 so let's say let me just out engineer it so that I can easily find the linear combination let me take let me take 3 times 3 times v1 plus plus 2 times 2 times v2 what is that going to be equal to that's going to be equal to the vector so 3 times 2 which is 6 plus 2 times 1 so this is the vector 8 the vector 8 and then I have the vector 3 times V 1 plus 2 times that so 8 7 right 3 plus 2 times 2 is 4 8 7 so if we were to just graph 8 7 in the traditional way we would go 1 2 3 4 5 6 7 8 and then we would go up 1 2 3 4 5 6 7 and we would have a vector I'm not going to draw out here but it would specify it would specify that point right there that would be this point if we if you view these as coordinates we will view that as the point eight seven right there maybe I'll write it like that that's the point eight seven if I wanted to draw the specter in standard position I would draw a vector that ends right there now we have this basis here this basis be represented by these two vectors this is V 1 and V 2 and what we want to do is represent this guy let's say that I have this vector let me call this vector let me call that vector a so vector a is equal to eight seven now we know that if we wanted to represent vector a as a linear combination of my basis vectors it's going to be three times v1 plus two times v2 so just given what we just saw in the earlier part of this video we can write that the vector a the vector a with respect to the basis B maybe I'll do it in the same color as a basis with respect to the basis B is equal to these weights on the basis vectors is equal to three and two now let's see if we can visually understand why this makes sense we're saying we're saying that in some new coordinate system where this vector can be represented as 3/2 and the way you think about the new coordinate system is in this old coordinate system we hashed out ones in the horizontal axis and we hashed out and that was our first coordinate and we hashed out ones in the vertical direction that was our second coordinate now in our new in our new system what's our first coordinate our first coordinate is going to be multiples is going to be multiples of V 1 this is V 1 or this is V 1 so it's multiples of V 1 so that's 1 times V 1 then if we do 2 times V 1 we're going to get over here 2 times V 1 would get us to what - will get us to four - four - three times V 1 would get us to 6 3 so let's see 1 2 2 3 4 5 6 7 8 so 6 and then 3 just like that and then four times v1 would get us to eight and four all right so you can imagine that what I'm drawing here this is kind of the axes the the first term axis generated by v1 so I could draw it let me do it in in this blue color so you could imagine it could imagine it like this just be a straight line just like that and then the coordinate tells me how many V ones do I have so I would hatch off the coordinate system like this instead of doing increments of one I'm going to do increments of v1 increments of v1 right just like that as you go 9 10 we're going to go up one more to 5 something like that right now the second coordinate tells you increments of v2 so this is our first increment of v2 then our second increment if we go to 4 it's going to be 4 2 just like that that's going to be 6 and 3 it's going to be just like that then it's going to be so 6 and 3 so it's going to look something like this so if you want to think of it as a bit of a well you should be thinking of as a coordinate system you can you can have this new skewed graph paper where any point you can now specify it as going in the v1 direction by some amount and then by going in the v2 direction by someone let me draw that as a bit of graph paper so I could draw I could draw another version of v2 just like that just all the multiples of v2 I could shift them like that I could do another one like this I could do another that one is a little bit not neat enough I could do it like that I could do I think you're getting the idea like let me make that a little bit neater this might have been useful to do it another tool and then I could do all the multiples of v1 like this I'm doing a graph paper right here so look something like this it would look something like this it would look something like this and so you can imagine the skewed graph paper if I did it all over the place with this kind of green and this blue so in our new coordinate system we're saying 3 2 so that means 3 times our first direction which is to be the v1 directions no longer the horizontal direction is the v1 direction so we go we're going one two and then we go three like that and then we're going to go two in the v2 direction so we're going to go one two in the v2 direction and so our point is going to be right there right you could imagine going like this you go three in the v1 direction and then you go one two and the v2 direction you get to our point or you could go kind of in your v2 direction and then your view one direction but either way you're going to get to your original point so that vector or that the position specified by the vector eight seven could just as easily be specified in our new coordinate system by the coordinates three two because we're saying three times v1 and then plus three times v1 it takes us in this direction we're going three notches in the v1 direction and then we're going to notches in the v2 direction and so that's why these are called that's why these are called coordinates coordinates you're literally saying how many how many spaces in the v1 direction to go and then how many spaces the v2 direction to go but this might this might I guess lead you to the obvious question why have we been using the coordinates before like I might have been saying all along I've been saying all along they have some vector I don't know let's say I have some vector lowercase B that is equal to I don't know let's say it's equal to I'll do it in r2 just cuz it's easy to visualize let's say it's equal to three minus one and if we you know if we were to graph it if we were to graph it it would look something like this we would go one two three and then we would go down one so it would look something like this it would specify this point but why have we been calling three and negative one coordinates why have we been calling three and minus one cord it's we've been doing it well before we learn linear algebra we call these coordinates all the way from when we first learned how to graph why are we calling those coordinates or how does this meaning of coordinates relate to these coordinates with respect to a basis well these are coordinates with respect to bases these are actually coordinates with respect to the standard basis if you imagine let's see the standard basis in r2 looks like this we could have e1 which is 1 0 and we have e 2 which is 0 1 these are just this is just the convention for the standard basis in r2 and so we could say if we call let's say s is equal to the set of e1 and e2 then we say that s is the standard standard basis basis for r2 and it's the standard basis because these two guys are orthogonal they're just this is 1 in the horizontal direction this is 1 in the vertical direction and any vector any vector in r2 let's say I have some vector let's say I have some vector X Y in r2 it's going to be equal to x times e 1 plus y times y times e2 so we could say that if you want to write if you want to write some vector X Y if you wanted to write it with respect to this standard basis right here with respect to the standard basis so I'm going to write it with respect to the standard basis it's going to be equal to the coordinates by the definition that we did earlier in this video of the basis vectors right there are these weights on our u ones and u 2 so it's going to be equal to well the weight there is X and the weight here is y so these coordinates that we've been talking about from the get-go these are definitely coordinates they're consistent with our definition of coordinates in this video but we can maybe be a little bit more precise we can now call them the coordinates coordinates with respect with respect to the standard basis and 'red with respect to the standard basis or we could call them we could call these right here we can call these the standard coordinates standard coordinates I just wanted to point this out this might be almost trivially simple or a bit obvious but I just want to show that our old usage of the words coordinates was not inconsistent with this new definition of coordinates as being the weights as being the weights on some some basis vectors because even in our old coordinates or the old way we use them these really were weights on our standard basis vectors