Stable general linear group over a field

Definition
Let $$K$$ be a field. The stable general linear group over $$K$$, denoted $$GL(K)$$, is defined in the following equivalent ways:


 * Define $$K^\omega$$ as the direct sum of countably many copies of $$K$$. $$GL(K)$$ is defined as the group of those linear automorphisms of $$K^\omega$$ that fix pointwise all but finitely many of the copies.
 * Consider the general linear groups $$GL_n(K)$$, with a natural inclusion map $$GL_n(K) \to GL_{n+1}(K)$$ that sends $$A$$ to the matrix with block description $$\begin{pmatrix}A & 0 \\ 0 & 1 \\\end{pmatrix}$$. $$GL(K)$$ is defined as the direct limit of this sequence of groups with homomorphisms.

Note that the stable general linear group is a proper subgroup of the general linear group of countable degree over a field.