Show that for a set of vectors to be linearly independent in a subfield $F$ of a larger field $G$, they must be linearly independent in $G$.

Question

$F$ is a subfield of a larger field $G$.

Let $ v_1, v_2,\ldots,v_m $ be vectors in $F^n $. Show that $ v_1, v_2,\ldots,v_m $ are linearly independent in $ F^n $ $iff$ they are linearly independent in $G^n$.

Edit:

I know that if I extend this set of vectors to a basis of $F^n$ and form a square matrix, this matrix is invertible over $F$, hence it's also invertible over $G$ since $F^n \subset G^n$,

however I'm failing to work the way the question is asking me to.

Apologies, I've been working on another longer question and just came back to this one. I've edited the question. — Peter C, Mar 14 '16 at 05:14

score 0 · Answer 1 · answered Mar 14 '16 at 06:01

Hint (direction 1): vectors in $F^n$ are vectors in $G^n$.

If no set of coefficients chosen from $G$ gives a relation, picking from a subset of $G$ can't do so either.

Hint (direction 2): $G$ has a basis and each of the $v_i \in F$ has a unique expansion in that basis.

If there were a relation among the $v_i$ over $F$ there would be a relation (the same one) among their unique expansions over $G$.

Show that for a set of vectors to be linearly independent in a subfield $F$ of a larger field $G$, they must be linearly independent in $G$.

1 Answers1