Define the entropy of a random variable $X$ by : $$H(X):=\sum_{x\in X(\Omega)}p(x)\log\left(\frac{1}{p(x)} \right), $$ with $0\log 0:=0$ and $p(x)=P(X=x)$.
A fundamental propriety satisfied by $H$ is : $$H(X)=\log |range(X)|, $$ if and only if $X$ is a uniform random variable.
Let $E$ be a finite-dimensional space vector over a finite field $\mathbb{F}$, then and $E^{*}$ the dual space. Then, if $X$ is a uniform variable over $E^{*}$ : $$H(X)=\log |range(X)|=\log |E^{*}|=\log |\mathbb{F}^{\text{dim}(E)}|)=\text{dim}(E)\log|\mathbb{F}|.$$
My question is : if $U,V$ are two linear subspaces of $E$ an $X,Y$ two uniform variables over $U$ and $V$, how do you prove that : $$H(X,Y)=\text{dim}(U+V)\log|\mathbb{F}|. $$
Note : In the following article
https://terrytao.wordpress.com/2017/03/01/special-cases-of-shannon-entropy/
Tao wrote that "the joint random variable $(X,Y)$ determines the random linear function $f$ $\color{red}{\text{on the union } U \cup V}$ on the two spaces, and $\color{red}{\text{thus by linearity on the Minkowski sum }U+V}$ as well; thus $(X,Y)$ is equivalent to the restriction of $f$ to $U+V.$ In particular, $H(X,Y) = \mathrm{dim}(U+V) \log |\mathbf{F}|$.",
but I don't really understand the 'red' argument.