Why Do Different Texts Present Bilinear Forms Differently?

[Math Amateur]

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

 

[Deveno]

I believe McInerney is in error.

 

[Math Amateur]

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