I am confused by the definition of the general linear group of a vector space. Let $V$ be a vector space of dimension $3$ and $W$ be a subspace of $V$ with dimension $2$. (For example, $V$ can be the three dimensional Euclidean space, and $W$ can be a plane in $R^3$.) We can observe that elements of $V$ and $W$ are going to be $3$-tuples.
When the general linear group of a vector space is defined, the size of matrix elements is given by the dimension of the vector space. So $GL(V,R)$ will have invertible real matrices of size $3\times3$ as elements. However, I am really confused when I am looking at $GL(W,R)$. Using the definition it seems its elements are invertible $2\times2$ real matrices. However, these matrices should be able to act on elements of $W$, so if $A$ is in $GL(W,R)$ and $w$ is in $W$ (note that $w$ is a vector in $R^3$). Then $A*w$ should be well defined, but it's not because $A$ is $2\times2$ and $w$ is $3\times1$.
I am pretty sure the elements of $GL(W,R)$ should be $3\times3$ matrices. However, how does this fulfill the definition of the general linear group of the finite dimensional vector space $W$? The dimension of $W$ is certainly not $3$. There is something I am misinterpreting, please any help would be appreciated. Thanks!