Linearly independent field homomorphisms.

[NoDoubts]

Should be simple, but can't figure out :)

Why is that , for a field K, the linear independence of field homomorphisms g1, ..., gn: K -> K
equivalent to the existence of elements a1, ..., an \in K such that the determinant

det| gi(aj)| != 0 (...so, just like in a case of linear independence of vectors).Thanks!

 

[NoDoubts]

I mean, one direction is clear i.e. if there exist n elements with the det != 0 then homomorphisms are clearly not linearly independent...

I don't see the other direction i.e. given n linearly independent homomorphisms why there exists n elements with the above property?

 

[NoDoubts]

guys, need help. this should be easy. pls let me know if anything is unclear.

the statement about different field homomorphisms being linearly independent is called Dedekind lemma. My question is why being linearly independent in this case implies existence of n elements a1, ..., an \in K such that determinant of g_i (a_j) is not zero.

thanks a lot!