Why Do Different Texts Present Bilinear Forms Differently?

In summary, the conversation is about a question regarding a proposition in Andrew McInerney's book on differential geometry. The proposition, which deals with the dual of a vector space and forms, is found to be stated differently in other books by Seymour Lipshutz and Bruce Cooperstein. After discussing the discrepancy, it is concluded that McInerney's proposition may be incorrect. The conversation ends with the person thanking Deveno for clarifying and stating that they will proceed with their reading.
  • #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
 

FAQ: Why Do Different Texts Present Bilinear Forms Differently?

What is a bilinear form?

A bilinear form is a mathematical function that takes in two vector inputs and produces a scalar output. It is a type of multilinear form, meaning it is linear in both of its inputs. In other words, if one input is multiplied by a constant, the output is also multiplied by that same constant.

How are bilinear forms represented mathematically?

Bilinear forms can be represented using matrices. The matrix entries correspond to the coefficients of the variables in the bilinear form. For example, a bilinear form with two variables may be represented by a 2x2 matrix.

What is the relationship between bilinear forms and inner products?

Bilinear forms are closely related to inner products, which are also functions that take in two vector inputs and produce a scalar output. In fact, inner products can be thought of as a special case of bilinear forms, where the bilinear form is symmetric and positive definite.

How are bilinear forms used in applications?

Bilinear forms have many applications in mathematics and physics. For example, they are used in optimization problems, where they can be used to represent constraints or objective functions. They are also used in geometry, where they can be used to find distances between points or angles between vectors.

What is the transpose of a bilinear form?

The transpose of a bilinear form is a new bilinear form that is obtained by switching the order of the inputs and taking the transpose of the matrix representation. In other words, if the original bilinear form is represented by a matrix A, the transpose is represented by the transpose of A.

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