Let $K\subseteq L$ be a field extension, and let $V$ be a $K$-vector space. The

**extension of $V$ by scalars in $L$**is the tensor product $E=V\otimes_KL$. I will prove that every $L$-vector spaced is obtained as some extension in this way and that $\dim_L(E)=\dim_K(V)$. Do you want to know more?