soft question - Should linear algebra and vector calculus from traditional courses be replaced with `geometric algebra`?

geometric algebra gives geometric meaning to linear algebra and much more. it can provide a coordinate free geometric interpretation of spaces. those who learn of it, tend to be dismayed they weren't taught physics in this framework. what are the pro's and con's of replacing linear algebra and vector calcunningulus with geometric algebra?

Good books on geometric algebra:

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  Geometric Algebra for Computer Science has exercises to self tests, and features prettyfied pictures for extra clarity.

  
  


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  Books by Hestenes have an incredible signal to noise ratio (extreme compression of information to Shannon Limit) and are definitely a must...

  
  


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  many introductions floating around, remember search for "geometric algebra" and not "algebraic geometry"

  
  

Also try to see the Pascal triangle structure of number of k vector basis blades in n dimensions!

Answer

For several years I have been teaching Clifford (geometric) algebra as part of the Vector Analysis Course for undergraduate physics majors in Ateneo de Manila University. I strictly use Cl_{n,0}, even for Special Relativity. 18-year old students do not complain how difficult geometric algebra is. They just learn the math and the geometric interpretations: geometric product, dot product, wedge product, cross product, exponentials of imaginary vectors for circular rotations, exponentials of vectors for hyperbolic rotations, etc. For linear algebra, I teach them how to rewrite simultaneous linear equations in vector form and use the wedge product to solve for the unknown parameters. All the properties of determinants are encoded in the wedge product of arbitrary number of vectors. Cylindrical and spherical coordinate systems are best taught in terms of exponential rotation operators, because students immediately see what are the axis of rotation, the angle of rotation, and the vector to be rotated. Vector calculus is also simpler because the gradient, divergence, and curl becomes part of a single spatial derivative operator which may act on scalars or vectors. This becomes very useful when we want to combine all the four Maxwell's equations into one. I think, the more one knows too much math such as tensors, spinors, and differential forms, the more it becomes difficult to understand geometric algebra. Geometric algebra is best taught to the little ones.