Let $K\subseteq L$ be a field extension, and let $V$ be a $K$-vector space. The <b>extension of $V$ by scalars in $L$</b> 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 $\dim_L(E)=\dim_K(V)$. <a href="https://blag.nullteilerfrei.de/2013/02/26/extension-by-scalars-and-dimension/#more-1390" class="more-link">Do you want to know more?</a>