Find the dimension of $V$ as a vector space over $\mathbb F$ Let $\mathbb F$ be a subfield of a field $\mathbb K$ satisfying  the condition that the dimension of $\mathbb K$ as a vector space over $\mathbb F$ is finite and equal to $r$. Let $V$ be a vector space of finite dimension $n>0$ over $\mathbb K$. Find the dimension of $V$ as a vector space over $\mathbb F$.
I am not sure how to even approach this problem. Looking around online I found $\dim_{\mathbb F}(V)=\dim_{\mathbb F}(\mathbb K)\dim_{\mathbb K}(V)$ for a field extension $\mathbb K/\mathbb F$. Where does this come from?

Let $\dim_K V = m$ and write $v\in V$ as $\displaystyle\sum_{i=1}^m a_i v_i$. Then write $a_i\in K$ as $\displaystyle\sum_{j=1}^nb_{ij}a_j$. Put them together to get

$$v = \sum_{i=1}^m\sum_{j=1}^nb_{ij}v_{ij}$$

where $\displaystyle a_iv_i = \sum_{i=1}^n b_{ij}v_{ij}$. The minimality of $mn$ is given by minimality of each of $m$ and $n$ in their respective settings.
