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I am reading Andrew McInerney's book: First Steps in Differential Geometry: Riemannian, Contact, Symplectic ...
I am currently focussed on Chapter 2: Linear Algebra Essentials ... and in particular I am studying Section 2.8 The Dual of A Vector Space, Forms and Pullbacks ...
I need help with a basic aspect of Proposition 2.8.14 ...
Proposition 2.8.14 reads as follows:https://www.physicsforums.com/attachments/5272I wanted some computational examples related to this proposition ... ... so I searched in the following books ...
Linear Algebra by Seymour Lipshutz (Schaum Series)
and
Advanced Linear Algebra by Bruce Cooperstein (CRC Press)... ... BUT ... ... I was confused by an apparent difference in the statement of the Proposition/Theorem ...The equivalent proposition/theorem in Lipshutz reads as follows:https://www.physicsforums.com/attachments/5273The equivalent proposition/theorem in Cooperstein reads as follows:View attachment 5274Now both Cooperstein and Lipshutz seem to have reversed the role of the $$w$$ and $$v$$ in McInerney's proposition ... that is, in their notation they seem to assert the following:
$$b(v,w) = v^T B w $$
Can someone please explain the apparent discrepancy ... ?
Help will be appreciated ...
Peter
I am currently focussed on Chapter 2: Linear Algebra Essentials ... and in particular I am studying Section 2.8 The Dual of A Vector Space, Forms and Pullbacks ...
I need help with a basic aspect of Proposition 2.8.14 ...
Proposition 2.8.14 reads as follows:https://www.physicsforums.com/attachments/5272I wanted some computational examples related to this proposition ... ... so I searched in the following books ...
Linear Algebra by Seymour Lipshutz (Schaum Series)
and
Advanced Linear Algebra by Bruce Cooperstein (CRC Press)... ... BUT ... ... I was confused by an apparent difference in the statement of the Proposition/Theorem ...The equivalent proposition/theorem in Lipshutz reads as follows:https://www.physicsforums.com/attachments/5273The equivalent proposition/theorem in Cooperstein reads as follows:View attachment 5274Now both Cooperstein and Lipshutz seem to have reversed the role of the $$w$$ and $$v$$ in McInerney's proposition ... that is, in their notation they seem to assert the following:
$$b(v,w) = v^T B w $$
Can someone please explain the apparent discrepancy ... ?
Help will be appreciated ...
Peter