2.2 An Outline of Geometric Algebra | Geometric Algebra for Physicists

Transcript

welcome to section 2.2 an outline of geometric algebra so clifford defined an algebra where an elements of any type would be added or multiplied together clifford by the way was a mathematician uh you probably have heard of him he's pretty famous he called this a geometric algebra which we actually now typically call clifford algebra geometric algebra refers to clifford algebra but with the intent of looking at its geometric properties and not really looking at as as a matrix notation elements of this algebra are called multi-vectors and they form a linear space scalars can be added to vectors bivectors and so on geometric algebra is a graded algebra and elements of the algebra can be divided into terms of different grade to express grade we typically use these angle bracket notations with a subscript and that subscript denotes the grade or like you could kind of think of it as a dimension of the individual term but we tend to use grade because dimension is typica is used for the overall size of a vector space or like a linear space but you can think of grade as being the dimension of the hyperplane that a term occupies and also an important thing to note is that when you're doing this to denote a scalar you can just remove this part and it's just assumed that it means a scalar and for the results that you see in this slide from these properties you will see these properties in action within the next two section episodes in planar geometric algebra and spatial geometric algebra these will be evident another thing that makes this uh geometric algebra useful as a tool is that you can divide by vectors which is given in this simple kind of proof right here there's not much else to say about that it's cool you can divide that's pretty neat and another thing is that proving that the square of a bi vector is negative in value which i have to tell you this like while this isn't necessarily a complex proof this negative sign i did not write down when i first was doing this proof and so when i got the positive sign i was like uh something is wrong and for some reason i just didn't realize that i just forgot this negative sign right here but anyway this should be a pretty straightforward proof there's not much else i need to say oh yeah thanks for watching this is just an outline of kind of what you should expect to see from geometric algebra as we explore in different dimensions the next episode will be section 2.3 on planar geometric algebra which is geometric algebra in two dimensions