Evaluating Linear Transformations Using a Basis

[sk57m]

@Bob Sorry...but I have nt really got it. Precisely, my doubt is that how do I ensure that changing the operations has no effect on the dimension?

[MathDoctorBob]

@sk57m Sorry about the confusion; for some reason, I thought we wanted the same dimension. The main point is that we can turn any set X in bijection with R^n into a real vector space of dimension n. That includes X = R^m, m gt 0.  - Bob

[sk57m]

is it possible to have any vector space structure on R^n other than the usual one(component wise addition and scalar multiplication)??

[MathDoctorBob]

@sk57m If we use a new addition and scalar multiplication satisfying the vector space axioms, choosing a basis gives an isomorphism to R^n with the usual vector space structure. The only field automorphism of R is the identity, so there is no analogue to conjugation.

Here's how to make new operations: Let A be a bijection from (R^n, usual) to (R^n, new). Define operations on (R^n, new) by v +' w = A( A^-1v + A^-1w ) and c*v = A(cA^-1v). Then the axioms should be verified. - Bob

[djd259th]

Great video, thanks!

[MathDoctorBob]

@djd259th You're welcome! - Bob