- #1
Hiero
- 322
- 68
It seems most people say that a vector is either contravariant or covariant. To me it seems like contra/covariance is a property of the components of a vector (with respect to some basis) and not of the vector itself.
Any basis {bi} has a reciprocal basis {bi} and any vector can be expressed with respect to either basis, ## v = v^i b_i = v_i b^i## (sums on i).
So then it seems a vector is just a vector and it’s the components which are either covariant or contravariant (wrt to the original basis)
One argument I heard is that the covariant vector somehow lives in some other (“dual”) space. This seems silly to me because I see no reason why the reciprocal basis can’t be taken to span the exact same vector space as the original basis.
What are your thoughts? Is there a good reason why I should consider contra/covariance to be a property of the vector itself?
Any basis {bi} has a reciprocal basis {bi} and any vector can be expressed with respect to either basis, ## v = v^i b_i = v_i b^i## (sums on i).
So then it seems a vector is just a vector and it’s the components which are either covariant or contravariant (wrt to the original basis)
One argument I heard is that the covariant vector somehow lives in some other (“dual”) space. This seems silly to me because I see no reason why the reciprocal basis can’t be taken to span the exact same vector space as the original basis.
What are your thoughts? Is there a good reason why I should consider contra/covariance to be a property of the vector itself?