General linear group of a subspace 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!
 A: As others have pointed out in the comments, $\DeclareMathOperator{\GL}{GL} \GL(V)$ is the set of invertible linear maps $V \to V$; only by choosing a basis do we get an isomorphism $\GL(V) \cong \GL_3(\mathbb{R})$.  Nonetheless, I think there is a way in which your question does make sense.  Given $\varphi \in \GL(W)$, $\varphi: W \to W$ is only defined on $W$, not on all of $V$, but there is a way to extend $\varphi$ to all of $V$.
Choose a basis $\mathcal{B} = \{e_1, e_2\}$ for $W$ and let $[\varphi]_\mathcal{B}$ denote the matrix representation of $\varphi$ with respect to $\mathcal{B}$. By the basis extension theorem, we can extend the basis $\mathcal{B} = \{e_1, e_2\}$ of $W$ to a basis $\mathcal{C} = \{e_1, e_2, e_3\}$ of $V$.  Define $\widetilde{\varphi} : V \to V$ by $\widetilde{\varphi}(e_i) = \varphi(e_i)$ for $i = 1,2$ and $\widetilde{\varphi}(e_3) = e_3$, i.e., $\widetilde{\varphi}$ acts as $\varphi$ on $W$ and $\widetilde{\varphi}$ fixes $e_3$.  Then the matrix representation of $\widetilde{\varphi}$ with respect to $\mathcal{C}$ is block-diagonal:
$$
[\widetilde{\varphi}]_\mathcal{C} =
\left(
\begin{array}{c|c}
[\varphi]_\mathcal{B} & 0 \\
\hline
0 & 1
\end{array}
\right) \, .
$$
So in some sense we have have embedded $\GL(W) \subseteq \GL(V)$, which in coordinates amounts to embedding $2 \times 2$ matrices in $3 \times 3$ matrices.  (More generally, this sort of block diagonal representation will arise whenever there is a $\varphi$-invariant subspace.)
