# Thread: Showing basis for Field?

1. ## Showing basis for Field?

let F be a field. we have
e1=(1 -1 1)
e2=(0 1 1)
e3=(1 0 1)
and need to show that {e1, e2, e3} is a basis for F^3

so far i've shown independence, but am stuck on the spanning bit.
please note that F^3 is not the mod3 field

thank you

2. For every

$(x_1,x_2,x_3)\in \mathbb{F}^3$

express:

$(x_1,x_2,x_3)=\lambda_1(1,-1,1)+\lambda_2(0,1,1)+\lambda_3(1,0,1)$

and prove that the corresponding system has solution on $\mathbb{F}$ for all $x_1,x_2,x_3$ .

Regards.

Fernando Revilla