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Actions of Field on Vector Space 
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#1
Nov1312, 11:20 PM

P: 597

Hi, Algebraists:
Say V is finitedimensional over F . Is there more than one way of defining the action of F on V (of course, satisfying the vector space axioms.) By different ways, I mean that the two actions are not equivariant. Thanks. 


#2
Nov1412, 04:19 PM

P: 345

But in general, there can be more than one possibility. For example, If V is a vector space over C (the complex numbers), and if a' denotes the conjugate of a complex number a, then we define a new action x of C upon V by a x v = a'v (where a'v is computed by the original action). It can be shown that V with this new action and the same addition as before satisfies the vector space axioms. 


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