Prove that tensor is second rank mixed

[Mcfly11]

Homework Statement

V^alpha and U^beta are both contravariant vectors, and obey the equation V^alpha=E^alpha_beta*U^beta. Show that E^alpha_beta is a mixed second rank tensor. (Note: I couldn't get the latex to work, my apologies for the ugly equations. E^alpha_beta means E with a superscript alpha, and subscript beta.)

Homework Equations

The transformation equation of mixed second rank tensors (kind of a bummer to type without the latex working for me...)

The Attempt at a Solution

The only solution I can come up with assumes that the equation is invariant under coordinate transformation...which doesn't seem valid to me. Basically, I transformed the two vectors V and U into 'barred' coordinates using the standard transformation for contravariant vectors, and substitute that into the equation given...then if the form of the equation is to remain invariant under coordinate transformations, it required that E follows the transformation properties of a second rank mixed tensor. I assume this is altogether invalid due to my assumption of invariance...can anyone offer me a starting point or a tip? Thanks!

 

[arkajad]

You reasoning is correct. You need to use two things: invariance (or covariance, if you wish), and arbitrariness of V and U. You did not mention the second condition and it was not explicitly stated in the problem. But evidently if that equation should hold only for, say, U=0, V=0, then you would not be able to say anything reasonable about E,