Linear independence of vectors over larger fields

Solution 1:

Yes. Extend the set $\{v_1,\ldots,v_k\}$ to a basis of $\mathbb{F}^n$ and put the basis vectors together to form a square matrix $A$. Then $A$ is invertible over $\mathbb{F}$. That is, $A^{-1}\in M_n(\mathbb{F})\subseteq M_n(\mathbb{G})$. So, $A$ is invertible over $\mathbb{G}$ and its first $k$ columns are linearly independent over $\mathbb{G}$.

However, do not confuse your question with the following one:

Suppose $\mathbb{F}$ is a subfield of $\mathbb{G}$ and the set $V$ is a vector space over each of $\mathbb{F}$ and $\mathbb{G}$. If $v_1,v_2,\ldots,v_k\in V$ are linearly independent over $\mathbb{F}$, are they necessarily linearly independent over $\mathbb{G}$?

The answer to this seemingly similar question is negative, as illustrated by the following counterexample: $V=\mathbb{C},\,\mathbb{F}=\mathbb{R},\,\mathbb{G}=\mathbb{C}$ and $\{v_1,v_2\}=\{1,\,i\}$. It is easy to see that $v_1$ and $v_2$ are linearly independent over $\mathbb{R}$ but linearly dependent over $\mathbb{C}$.