Proving Non-Decomposability of Wedge Product in Higher Dimensional Vector Spaces

In summary, facenian asked for a nicer way to tackle the problem of proving that e_1\wedge e_2 + e_3\wedge e_4 is not decomposable when the dimension of the vector space is greater than 3 and e_i are basis vectors. Tiny-tim suggested expressing a and b in terms of the basis and then concluding that a must be null. Facenian then clarified the method and tiny-tim confirmed it. In the end, tiny-tim's method was deemed better than facenian's.
  • #1
facenian
436
25
I have this problem(from Tensor Analysis on Manyfolds by Bishop and Goldberg): prove that
[itex]e_1^ e_2 + e_3^e_4[/itex] is not decomposable when the dimension of the vector space is greater than 3 and e_i are basis vectors.
I solved it by mounting a set of 6 equations with 8 unknows and studying the different posibilities cheking that each one is not solvable.
Is there any nicer way to tackle this problem? if so please let me know
 
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  • #2
hi facenian! :smile:

(use "\wedge" in latex :wink:)
facenian said:
I have this problem(from Tensor Analysis on Manyfolds by Bishop and Goldberg): prove that
[itex]e_1\wedge e_2 + e_3\wedge e_4[/itex] is not decomposable when the dimension of the vector space is greater than 3 and e_i are basis vectors.
I solved it by mounting a set of 6 equations with 8 unknows and studying the different posibilities cheking that each one is not solvable.
Is there any nicer way to tackle this problem? if so please let me know

you need to prove that it cannot equal [itex]a\wedge b[/itex] where a and b are 1-forms …

so express a and b in terms of the basis :wink:
 
  • #3
tiny-tim said:
hi facenian! :smile:

(use "\wedge" in latex :wink:)


you need to prove that it cannot equal [itex]a\wedge b[/itex] where a and b are 1-forms …

so express a and b in terms of the basis :wink:

helo tiny-tim, thanks for your prompt response and yes I did what you suggested and it led me to what I explained
 
  • #4
how about [itex]a\wedge (e_1\wedge e_2 + e_3\wedge e_4)[/itex] ? :wink:
 
  • #5
tiny-tim said:
how about [itex]a\wedge (e_1\wedge e_2 + e_3\wedge e_4)[/itex] ? :wink:

you mean, let [itex]a=\sum_{i<j} x_{ij} e_i\wedge e_j[/itex] and then conclude tha [itex]a[/itex] must be null? Please let me know if that's what you meant and/or if I'm correct
 
  • #6
hi facenian! :smile:

no, I'm using the same a as before (in a∧b, which you're trying to prove it isn't)

so let a = ∑i xiei :wink:
 
  • #7
I'm sorry I did not explained it correctly I should have said:

you mean, let [itex]a=\sum_i x_{i} e_i[/itex] and then conclude tha [itex]a[/itex] must be null because we are left with a linear conbination of basic vectors of the form [itex] \sum x_i e_i\wedge e_j\wedge e_k=0[/itex] .Please let me know if that's what you meant and/or if I'm correct
 
Last edited:
  • #8
facenian said:
you mean, let [itex]a=\sum_i x_{i} e_i[/itex] and then conclude tha [itex]a[/itex] must be null because we are left with a linear conbination of basic vectors of the form [itex] \sum x_i e_i\wedge e_j\wedge e_k=0[/itex] …

… which has to be 0, because a ∧ (a ∧ b) = 0

yes :smile:
 
  • #9
thank you very much tiny-tim your method is much better than mine!
 

1. What is the wedge product in mathematics?

The wedge product is a binary operation in mathematics that is used to combine two vectors into a new vector. It is denoted by the symbol ∧ and is also called the exterior product or the outer product.

2. How is the wedge product different from the dot product?

The wedge product is different from the dot product in several ways. Firstly, while the dot product results in a scalar quantity, the wedge product results in a vector quantity. Secondly, the dot product is commutative, but the wedge product is not. Lastly, the wedge product is used to calculate the area of a parallelogram formed by two vectors, while the dot product is used to calculate the projection of one vector onto another.

3. What are some applications of the wedge product?

The wedge product has many applications in mathematics and physics. In geometry, it is used to calculate the volume of a parallelepiped formed by three vectors. In physics, it is used to describe the orientation of a rigid body in three-dimensional space. It is also used in differential geometry to define the exterior derivative and the wedge product of differential forms.

4. Can the wedge product be generalized to higher dimensions?

Yes, the wedge product can be generalized to higher dimensions. In three-dimensional space, the wedge product results in a vector. In higher dimensions, it results in a multivector, which is a combination of scalars, vectors, bivectors, trivectors, etc. The concept of the wedge product can be extended to any number of dimensions, making it a powerful tool in higher-dimensional mathematics.

5. How is the wedge product related to the cross product?

The wedge product and the cross product are closely related, but they are not the same. Both operations take two vectors as inputs and result in a new vector. However, the cross product is only defined in three-dimensional space, while the wedge product can be defined in any number of dimensions. Also, the cross product is anti-commutative, while the wedge product is not. However, in three dimensions, the cross product can be expressed as a special case of the wedge product.

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