- #1
losiu99
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Homework Statement
Let E be a vector space of finite dimension over [tex]\Gamma[/tex] (char 0), and [tex]F\colon L(E,E)\rightarrow \Gamma[/tex] satisfies
(1) [tex]F(\phi \circ \psi)=F(\phi)F(\psi)[/tex]
(2) [tex]F(\hbox{id})=1[/tex]
Prove F can be expressed as a function of determinant, [tex]F(\phi)=f(\hbox{det}\phi)[/tex].
Homework Equations
Hint: Let [tex]e_\nu[/tex] be a basis and define
[tex]\psi_{ij}e_{\nu}=\begin{cases}e_\nu & \text{if } \nu\neq i\\e_i+\lambda e_j & \text{if }\nu = i\end{cases}[/tex]
[tex]\phi_i e_\nu =\begin{cases}e_\nu & \text{if } \nu \neq i \\ \lambda e_\nu & \text{if }\nu = i\end{cases}[/tex]
Show that [tex]F(\psi_{ij})=1[/tex] and that [tex]F(\phi_j)[/tex] doesn't depend on i.
The Attempt at a Solution
I have proved facts in the hint, but I cannot clearly see how to carry on. I guess it's about using this two families of functions to transform functions having the same determinant without changing the value of F to the point we can conclude F's values are equal. Unfortunetely, I fail miserably trying to accomplish this.
I appreciate any help.