Studying Geometric Algebra: Degenerate & Nondegenerate Forms Explained

In summary, the person is having difficulty understanding degenerate symmetric bilinear forms and seeks an example and brief explanation. They also mention the opposite, nondegenerate nonsymmetric bilinear forms, and hope that one explanation will be enough to understand. They apologize for not understanding everything in self study and mention using Google to find more information.
  • #1
Gott_ist_tot
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I'm trying to study geometric algebra using Artin's book and am having some difficulty with what degenerate symmetric bilinear forms would be like. Does someone know of an example and brief explanation. Also, the opposite being "nondegenerate nonsymmetric bilinear form" would help me out. If I have one explanation, that will probably help me enough though. I apologize, but sometimes I don't get everything in self study. Thanks!
 
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  • #2
Google is your friend

http://en.wikipedia.org/wiki/Degenerate_form

Popped up as the second or third hit

Similarly for non-symmetric bilinear forms

http://en.wikipedia.org/wiki/Bilinear_form
A bilinear form is symmetric if and only if the maps B1,B2: V -> V* are equal, and skew-symmetric if and only if they are negatives of one another. If char(F) ≠ 2 then one can always decompose a bilinear form into a symmetric and a skew-symmetric part as follows
where you can find the definitions of the Bi earlier in the article
 

Related to Studying Geometric Algebra: Degenerate & Nondegenerate Forms Explained

1. What is geometric algebra?

Geometric algebra is a mathematical framework that extends traditional vector algebra to include both scalar and vector quantities. It uses a combination of geometric and algebraic operations to describe geometric objects and their transformations in a concise and intuitive way.

2. What are degenerate and nondegenerate forms in geometric algebra?

Degenerate forms in geometric algebra refer to quantities that have zero magnitude or are parallel to other quantities, resulting in a loss of information and making them less useful for describing geometric objects. Nondegenerate forms, on the other hand, refer to quantities that have non-zero magnitude and are not parallel to others, making them more useful for describing geometric objects.

3. Why is it important to understand degenerate and nondegenerate forms in geometric algebra?

Understanding degenerate and nondegenerate forms is important because it allows us to distinguish between useful and useless quantities in geometric algebra. By eliminating degenerate quantities, we can simplify calculations and better describe geometric objects and their transformations.

4. What are some real-world applications of studying degenerate and nondegenerate forms in geometric algebra?

Geometric algebra has many real-world applications, including computer graphics, robotics, physics, and engineering. By understanding degenerate and nondegenerate forms, we can more accurately model and analyze physical systems and create more efficient algorithms for solving complex problems.

5. How can I learn more about studying geometric algebra and its degenerate and nondegenerate forms?

There are many resources available for learning about geometric algebra, including textbooks, online tutorials, and courses. You can also join online communities and forums where you can ask questions and discuss different aspects of geometric algebra with other scientists and mathematicians.

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