SUMMARY
The discussion centers on the isomorphism between the C-tensor product of C and C, and the R-tensor product of C and C as Q-modules. It is established that while the former is isomorphic to R² and the latter to R⁴ over R, both vector spaces are infinite dimensional over Q. The isomorphism holds due to the equality of the cardinality of their bases, which is the cardinality of the Continuum. The tensor product serves to extend the field of scalars, transforming a real vector space of dimension n into a complex vector space of dimension n and a real vector space of dimension 2n.
PREREQUISITES
- Understanding of tensor products in linear algebra
- Familiarity with vector spaces over different fields (R, C, Q)
- Knowledge of cardinality and dimensions of vector spaces
- Basic concepts of isomorphism in linear algebra
NEXT STEPS
- Study the properties of tensor products in linear algebra
- Explore the concept of vector spaces over different fields, focusing on R, C, and Q
- Learn about cardinality and its implications in vector space theory
- Investigate isomorphisms in the context of infinite dimensional spaces
USEFUL FOR
Mathematicians, students of linear algebra, and anyone interested in advanced topics related to vector spaces and tensor products.