Let $E$ and $F$ be two subspaces of $\mathbb{R}^n$, and let $$G = \{\begin{pmatrix} X \\ Y \end{pmatrix}\in \mathbb R^{2n} \mid X+Y \in E, Y \in F\}$$. I am trying to calculate the dimension of $G$, and I suspect that $\text{dim } G = \text{dim } E + \text{dim } F$, but I don't see how to justify it. Thanks if someone wants to help me.
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