Decomposable Tensors Problem: V Vector Space Dim ≤ 3

[Canavar]

Problem:
V a vector space with dimV \le 3, then every homogeneous element in \Lambda(V) is decomposable.


So, this exercise doesn't sound very difficult. My problem is, that i don't know the definition of homogeneous and decomposable. Can you please help me?
Thank you

 

[fzero]

A homogeneous element of \Lambda(V) is a form of definite degree, p. A decomposable form is one that can be written as a single exterior product of two forms of lower degree \omega = \mu \wedge \nu.

 

[Canavar]

Hello,

Thank you for your help!
are these examples correct?:
For dimV=2 We have \Lambda(V) = IK \times T^1(V) \times T^2(V).

-Then a homogeneous elm. would be all elm. in v\in \IK or v\in T^1(V) or v\in T^2(V).
Where T is the Alternator.

-Is it allowed to use the single exterior product of two form of equal degree, i.e. v=\mü \wedge \tau with v,\mü,\tau \in T^1(V) for instance?


Thanks

 

[fzero]

Hello,

Thank you for your help!
are these examples correct?:
For dimV=2 We have \Lambda(V) = IK \times T^1(V) \times T^2(V).

-Then a homogeneous elm. would be all elm. in v\in \IK or v\in T^1(V) or v\in T^2(V).
Where T is the Alternator.

Yes a homogeneous element would be in T^p(V) for some p.

-Is it allowed to use the single exterior product of two form of equal degree, i.e. v=\mü \wedge \tau with v,\mü,\tau \in T^1(V) for instance?

Yes it is possible that the decomposition involves forms of equal degree. However, in your example, if \mu, \tau \in T^1(V), then v= \mu \wedge\tau\in T^2(V).