I was just wondering whether anyone knows an answer to the following:
Suppose that ${\mathbb F}$ is a subfield of a field ${\mathbb G}$ and that $v_1,\ldots ,v_k$ are linearly independent vectors in ${\mathbb F}^n$ (over $\mathbb F$). Is it necessarily true that $v_1,\ldots ,v_k$ are also linearly independent when considered as vectors in ${\mathbb G}^n$ (over $\mathbb G$})?
Any help would be much appreciated. Thanks!
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 $\mathbb{C}$.
