Question

Let V be a finite dimensional real vector space and W be an n-dimensional sub-vector space in V. If a set Gr_(n)(V)={W⊂V} exists, create a manifold structure on Gr_(n)(V) of dimension n(dim(V)-n) and use it to prove there is a projective space when n=1, stating any theorems used along the way.

Name: let v be a finite dimensional real vector space and w be an n dimensional sub vector space in v if a set grnvwv exists create a manifold structure on grnv of dimension ndimv n and use it t 33715
Uploaded: 2022-06-02T11:28:41-08:00
Duration: 9 min 48 s
Channel: Joel Mueller
Description: let v be a finite dimensional real vector space and w be an n dimensional sub vector space in v if a set grnvwv exists create a manifold structure on grnv of dimension ndimv n and use it t 33715

          Let V be a finite dimensional real vector space and W be an n-dimensional sub-vector space in V. If a set Gr_(n)(V)={W⊂V} exists, create a manifold structure on Gr_(n)(V) of dimension n(dim(V)-n) and use it to prove there is a projective space when n=1, stating any theorems used along the way.

Added by Robert P.

Computer Science and Information Technology

Trishna Knowledge Systems 2018 Edition

Instant Answer

Solved by Expert Joel Mueller

06/02/2022

Step 1

To construct a manifold structure on \( Gr_n(V) \), we need to show that it can be given the structure of a smooth manifold. Show more…

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Let V be a finite dimensional real vector space and W be an n-dimensional sub-vector space in V. If a set Gr_(n)(V)={W⊂V} exists, create a manifold structure on Gr_(n)(V) of dimension n(dim(V)-n) and use it to prove there is a projective space when n=1, stating any theorems used along the way.