F is a field, K is a sub field of F , There is a subspace over the field K (and therefore F), and there is A=(a1,a2,....,an) vectors that belongs to the subspace V

Prove or disprove the next to claims

1-)If A is linearly independent over the field K , then A is linearly independent over the field F too.

2-)If A is linearly independent over the field F, then A is linearly independent over the field K too.

So, the second claim is wrong, and I gave a counter-example, the finite fields Z2 and Z3.

And I think the first claim is right because every vector that is in K, it is in F too. That's why if it is linearly independent in K then it should be in F too, F wouldn't make a difference for those vectors, however, this is not a proof, so How should I prove the first claim? Maybe it's not even correct, but I couldn't find a counter-example for that.

Appreciate if someone can help me proving this.

By the way, how should I do the prove\disprove question? Because there are a lot of claims where you think it's right , and you ''prove'' it and you find out it's wrong, so every time I do these kind of questions I should try to disprove it first right?