Why Do Different Texts Present Bilinear Forms Differently?

  • #1
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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 \(\displaystyle w\) and \(\displaystyle v\) in McInerney's proposition ... that is, in their notation they seem to assert the following:

\(\displaystyle b(v,w) = v^T B w \)
Can someone please explain the apparent discrepancy ... ?

Help will be appreciated ...

Peter
 
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  • #2
I believe McInerney is in error.
 
  • #3
Deveno said:
I believe McInerney is in error.


Thanks so much, Deveno ...

Given what you have said I will alter my text appropriately and read on ...

Thanks again,

Peter
 
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