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The determinant story of "distortion of unit volume" only makes sense AFAICT if you're in R^n. What does the determinant mean when you're considering linear fu
by bollu 8y ago
The determinant story of "distortion of unit volume" only makes sense AFAICT if you're in R^n. What does the determinant mean when you're considering
linear functions from 2^n to 3^n? There's no geometric interpretation I'm aware of for this case; indeed, I doubt there is one, due to the inherent nature of finite fields.
- Koshkin 8y ago> linear functions from 2^n to 3^n Say what?
- nsajko 8y ago2^n is {0, 1}^n, the space of n-tuples with elements from the smallest field. The field is also known as GF(2), Z/2Z, or F₂.
- ajkjk 8y agoA first-time student in linear algebra definitely does not need to hear anything about finite fields, though. I don't think anyone outside of certain mathematics subfields needs to.
- skrealing 8y agoGiven a vector space V, form its top wedge power. This is one-dimensional. And endomorphism of V induces an endomorphism of the top wedge power. But an endomorphism of a 1-dimensional vector space is a number. This number is the determinant. This is probably the correct generalization of the volume definition.
- devereaux 8y agoGeometric interpretations have a place in learning: until one is ready to understand another interpretation that is more complete, better, yet more complicated. You don't start kids with complex numbers until they can handle reals. You don't even start negative numbers until they can handle positive numbers.