Covector applied to a tensor product
From Maths
Contents
Definition
Given two vector spaces, [ilmath](V,F)[/ilmath] and [ilmath](W,F)[/ilmath] and a covector in the dual space of [ilmath]V[/ilmath], [ilmath]f^*\in V^*[/ilmath], with:
- [math]f^*:V\rightarrow F[/math]
We can define a map, denoted [ilmath]f^*:V\otimes W\rightarrow W[/ilmath][Note 1] as follows:
- [math]f^*:V\otimes W\rightarrow W[/math] by [ilmath]f^*(\sum^k_{i=1}v_i\otimes w_i)=\sum^k_{i=1}f^*(v_i)w_i[/ilmath][1]
Proof of claims
Claim: This is a linear map
Simple, just do [ilmath]f^*(\alpha(v_1\otimes w_1)+\beta(v_2\otimes w_2))[/ilmath] and find this gives the expected result
Claim: This is well defined
Given two representations for the same tensor products, we must show that [ilmath]f^*[/ilmath] of them both is well defined. See p48 LAVEP
Notes
- ↑ This isn't ambiguous because if I write [ilmath]f^*(v\otimes w)[/ilmath] it is clear I am talking about the tensor one, where as [ilmath]f^*(v)[/ilmath] is clearly about the usual covector one. The types of the variables at play remove the ambiguity
References
- ↑ Linear Algebra via Exterior Products - Sergei Winitzki
