To calculate the dimension of a vector space

Question

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.

Answer 1

In my opinion, you already answered your question in the comments.

Let $f:G\to E\times F$ be a linear map defined by $$f\begin{pmatrix}X\\Y\end{pmatrix}=(X+Y,Y).$$ Let $f^{-1}: E\times F\to G$ be a linear map defined by $$f^{-1}(W,Z)=\begin{pmatrix}W-Z\\Z\end{pmatrix}.$$ We can show that $$f^{-1}\circ f\begin{pmatrix}X\\Y\end{pmatrix}=f^{-1}(X+Y,Y)=\begin{pmatrix}(X+Y)-Y\\Y\end{pmatrix}=\begin{pmatrix}X\\Y\end{pmatrix};$$ $$f\circ f^{-1}(W,Z)=f\begin{pmatrix}W-Z\\Z\end{pmatrix}=\left((W-Z)+Z,Z\right)=(W,Z).$$ Thus, we have $f^{-1}\circ f=\operatorname{Id}_G$ and $f\circ f^{-1}=\operatorname{Id}_{E\times F}$, which means that $f:G\to E\times F$ is an isomorphism. Therefore, $$\dim(G)=\dim( E\times F)=\dim(E)+\dim(F).$$