- #1

Phinrich

- 82

- 14

- TL;DR Summary
- Hi All

Given a Vector Space V (of dimension n) and its Dual V* (also having dimension n) can we mathematically combine these vector spaces to form a Vector Space of Dimension 2n ? I dont believe the Direct Sum will do because I think to do a Direct Sum we will end up with a Direct Sum still having dimension n and not 2n. I stand to be corrected.

Any help would be greatly appreciated.

Given that the Set of 1-Forms is a Vector Space distinct from, but complimentary to, the Linear Vector Space of Vectors. And given that there is an Isomorphism between the linear space of vectors and the dual vector space of 1-forms, does it make mathematical sense to combine a vector space and its dual space of 1-forms to produce a new space having dimension equal to the sum of the dimension of the vector space plus the dimension of its dual? I believe that if we define V as {v1, v2, 0} and a second vector space W = {0,0,w1} then we can form the Direct Sum Space of the two as Y = {v1, v2, w1}. However since the components of V = {v1, v2,0} and the components of V* = {v1*,v2*,0} there is no way to form a Direct Sum and still satisfy the rules for a Direct Sum. So, in component form, I want to take {v1,v2,0} PLUS {w1,w2,0} to produce Y={v1,v2,0,w1,w2,0}.

Hope this makes sense to someone.

Thanks

Paul

Hope this makes sense to someone.

Thanks

Paul