Prove generalisation of the Tower Law

Question

I need to prove the following generalisation of the Tower Law:

Let $L/K$ be an extension of fields, and $V$ a non-zero vector space over $L$. Then $V$ is finite-dimensional over $K$ if and only if $V$ is finite-dimensional over $L$ and the extension L/K is finite.

However, I don't understand how the proof would be different from the main Tower Law. How do I alter it to prove this version?

Answer 1

if ${v_1,...v_n}$ spans V over K,its spans V over L,so V is finite dimensional over L,L isomorph to $Lv_1\in V$ so $L/K$ is finite. for converse {v_1,...v_n} span V over L and {l_1,...,l_m} span L over K,then it is easy to proof {l_1v_1,...,l_1v_n,...,l_mv_n} span V over K