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Modules are "simpler" than vectors because they have fewer axioms, but they are also much harder to understand. For example, not all modules have a basis, whic
by jackpirate 7y ago
Modules are "simpler" than vectors because they have fewer axioms, but they are also much harder to understand. For example, not all modules have a basis, which can make them much harder to work with.
For background on the math, see: https://math.stackexchange.com/questions/137442/a-module-without-a-basis https://math.stackexchange.com/questions/137442/a-module-wit...
- hzhou321 7y agoGood luck explaining "simpler" with modules and vectors :). Simple is defined as not to inter-wine. To understand an axiom is to understand how it "inter-wine" with other axioms to prove certain results. So fewer axioms necessarily results in more interwines, ie complex. I think here we are switching the subjects: from axiom itself to the results that we want to prove. If we focus on the simplicity of proving the results, the simplicity of axioms are irrelevant.
- F-0X 7y agoInteresting analogy, but it's a little off. The main reason modules are interesting is not as a generalisation of vector spaces, but because they are helpful in studying rings. Kernels of ring homomorphisms are ideals, which in general are not subrings, but they are modules - and of course every ring is a module over itself. So to study a ring R it pays off to instead study R-modules, since working with them is... you guessed it! Simpler.