# Stable general linear group over a field

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.