1.1 Vector Spaces | Geometric Algebra for Physicists

Transcript

hello um i decided that i would start a quick series that teaches everything within the textbook geometric algebra for physicists i've condensed the book into notes and then i've decided to take these notes and make presentations to teach other people it will be relatively simple and maybe oversimplified and i might get some things wrong the main reason that i'm doing this is to reinforce the learning myself but i thought that this could also maybe help somebody else so first we're going to start off with section 1.1 which is on vector spaces specifically linear vector spaces now here are the properties that you would find within it within a vector space additions commutative additions associative every vector has a negative inverse and everything else these are all axiomatic statements that you can prove relatively easily thus i will not prove them because it's kind of pointless to do so and then this leads us to our first definition which is a subspace definition basically saying that if there are two vector spaces and one vector space is contained within another one which you can kind of visualize like this if this is vector space v if there's another vector space within it u then all of the elements within u will be within v and so u is a subspace of v and then you can also call two vector spaces isomorphic if you can place uh them into a one to one which is an injective correspondence which preserves the sums and also there's a an injective correspondence between scalars this can kind of be viewed as having two separate two separate let's see vector spaces and say you have say like this isn't actually a thing i'm just using this as a visualization tool say it has like four sections in it and say this one has uh four or more this could have more sections but for every section here you can map it over to another section over here [Music] so for every section there is a correspondence and so these would become equivalent vector spaces here so you just call them isomorphic then another important thing which this is a definition from linear algebra which is that a vector b is a linear combination of vector a through vector n if there are scalars that can be found that can basically transform vectors a into vector b one kind of example of this is say uh like if you just like ignore up to the a the vectors a sub n and you just say a vector b is like this long vector right here this would be vector b let's just say that you have another vector that's like half the length and say its vector vector a well you can represent this as 2 times 2 times the vector a and so that that leads to vector b being 2 times vector a which is you have a scalar here such that a sub 1 or basically just a is equal to b so this is just a basic representation of what a linear combination is [Music] now for linear dependence also a definition from linear algebra uh you have a set of a um to a sub n vectors and they are linearly dependent if and only if there are scalars which are not all zero that can be found such that these these vectors all sum to zero with their respective scalars and then you can also basically come up with the definition for linearly independent a set is linear linearly independent if the only way to represent it as being equal to zero is if all the scalars are equal to zero basically that means uh basically for linear dependence that means that if you have like at some point scalars times certain vectors are going to cancel cancel out so let's just like uh let's just uh basically say if there's linearly if there's a set of vectors here [Music] if there's a set of vectors and then the vec the only vectors here are like one and two and then the second vector in it is two and four and this is pretty easy to show that it's linearly independent because you can represent this vector here um you can represent 2 for this vector as being equal to uh hang on to being one over two times the first vector which is basically a rep representation of this basically meaning this vector is not technically unique as it can be represented by a scalar times this first vector [Music] which as you can you can also show that if you just had a set with this vector in it that would be a linearly independent set because the only way for you to represent this as being zero equal to zero is if you have a zero scalar out front here but with this thing you could just have this this plus one half or plus a negative one half times this and that would equal zero without having the scalars all be zero [Music] now a set a sub 1 through a sub n vectors spans a vector space u if and only if every element of that vector space u is a can be expressed as a linear combination of that set which basically says if you have two vec like say we have like a a plane which goes out through here and then this is our vector space u if you have like a vector here and a vector here i don't know let's just call this one vector y and vector x well then any combination of these you can like if you have like a very big a larger scaler times this x you can create some new uh some new vector x prime here that goes all the way here and then if you vary the amount that you add this get uh you vary the scalar on this and then you add this like in some form lambda x plus mu y if you just like vary these you can just bring you can kind of like bring all vectors up to here and extend this vector here and then it'll just eventually cover all the space and so the linear combination of these two vectors spans this entire space that is what the definition and so this you would say the set of these like these two vectors here which you could say like maybe are the unit vectors and so this one would be one zero that looks like a six and then this one would be zero one any sort of scalar combination of these would end up uh representing the entire space [Music] and then basically a basis for that is a set that spans you but it's basically only made up of vector vectors like uh so like if if we go if we thought of uh the representation i had for the span you have two vectors here well you could also like if you had a vector here if you had a third vector that would still span all of this because you could still represent it all as a as a linear combination of these three vectors but this third vector right here can be represented as a linear combination of these two vectors which means that it's not linearly independent and so getting rid of this third vector and looking at the example that i showed in the last slide you can see that this would be a basis because this is the smallest set that you can have that will span you and then all of the aforementioned axioms and definitions i've given are sufficient to prove the basis theorem which states that all bases of a vector space have the same number of elements which basically means that any basis will have the same amount of a unit vec or vectors in it uh and so i came up oh yeah this is all called dimension but i came up uh like real quick so like forgive me if the proof isn't complete or like if there's a problem with it i i literally like last second wrote down this proof before i came here basically what it what it does is uh that it says it's like assume that you have uh basically assume that you have some uh vector space u and then you have some amount some amount of of vectors like uh and then this vector could could exist in the set or not if but if you have this q right here uh be greater than k which is means that they're they're say uh k for this instance would be uh equal to two and so if this vector existed that means q is equal to three basically i'm saying like if q is greater than the dimension of the vector space then this must be a linear combination which this this vector right here must be a linear combination of these two which means that it's not independent which violates the definition of a basis and so that case is not possible for it to be a basis for this u but if q is equal to k so if this vector didn't exist and this uh set right here happened to have some vectors that were within the same lines here uh then this basically says that these are still linearly independent and so and remembering that these span u by def by the definition of u basically existing those two combine to basically say that this must be a another basis for you and this might not be have the same vectors but it there are two two vectors in it and so there must be two vectors for every basis in u and you basically this is a generalized proof saying that if there are k if u is of k dimension then there must be k elements uh within the basis i don't know if that made sense and i hope it did um so yeah this is just uh but the first chapter uh 1.1 and next lesson i will go into 1.2 which is the scalar product keep in mind that the first chapter from geometric algebra for physicists is written uh very basically and only introduces uh old concepts because it's a review of previous previous concepts and so you don't really start getting into geometric algebra until chapter 2.