Linear algebra, field morphisms and linear independence

[mariang]

Homework Statement

Let f1,f2, ..., fn : K -> L be field morphisms. We know that fi != fj when i != j, for any i and j = {1,...,n}. Prove that f1,f2, ..., fn are linear independent / K.

Homework Equations

f1, ..., fn are field morphisms => Ker (fi) = 0 (injective)

The Attempt at a Solution

I tried to use the linearity and the injectivity but i got stuck.

 

[fresh_42]

Homework Statement

Let f1,f2, ..., fn : K -> L be field morphisms. We know that fi != fj when i != j, for any i and j = {1,...,n}. Prove that f1,f2, ..., fn are linear independent / K.

Homework Equations

f1, ..., fn are field morphisms => Ker (fi) = 0 (injective)

The Attempt at a Solution

I tried to use the linearity and the injectivity but i got stuck.

In such cases it is often helpful to start with $n=1$ and $n=2$ to see how possible arguments work. From there one can either proceed by a general $n$ or per induction. I assume we also have to require $f_i \neq 0\,.$