The general linear group, denoted $\operatorname{GL}(n, F)$ or simply $\operatorname{GL}(n)$ when the field $F$ is understood from context, is the set of all invertible $n \times n$ matrices with entries in a field $F$, along with the operation of matrix multiplication. ## Key Properties 1. **Invertibility**: A matrix $A \in \operatorname{GL}(n, F)$ must have an inverse, denoted $A^{-1}$, such that $A A^{-1} = A^{-1} A = I$, where $I$ is the identity matrix. In practical terms, this means the determinant of $A$ must be nonzero (i.e., $\det(A) \neq 0$). 2. **Matrix Multiplication**: The group operation in $\operatorname{GL}(n, F)$ is matrix multiplication. Under this operation, $\operatorname{GL}(n, F)$ is closed, associative, has an identity element (the identity matrix $I$), and every element has an inverse. 3. **Non-commutative Structure**: For $n \geq 2$, $\operatorname{GL}(n, F)$ is generally non-abelian, meaning that matrix multiplication does not necessarily commute; $AB \neq BA$ for matrices $A$ and $B$ in $\operatorname{GL}(n, F)$. ## Subgroups and Special Cases - **Special Linear Group**: The subgroup of matrices with determinant 1 is called the special linear group, denoted $\operatorname{SL}(n, F)$. This group is important in geometry and physics, as it represents transformations that preserve volume. - **Orthogonal and Unitary Groups**: When $F = \mathbb{R}$ or $F = \mathbb{C}$ (the field of real or complex numbers), subgroups of $\operatorname{GL}(n, F)$ can be defined to preserve certain properties, such as the orthogonal group $\operatorname{O}(n)$ (preserving lengths and angles) and the unitary group $\operatorname{U}(n)$ (preserving complex inner products). ## Significance The general linear group plays a fundamental role in linear algebra, geometry, and various areas of mathematics and physics, as it describes all possible linear transformations on an $n$-dimensional vector space over $F$ that can be reversed (i.e., are invertible). This invertibility is crucial because it means every transformation in $\operatorname{GL}(n, F)$ can be “undone,” making it a very robust structure for studying symmetry and transformation properties in higher-dimensional spaces.