General Linear Group
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Theorem
Let $K$ be a field.
The set of all invertible order-$n$ square matrices over $K$ is a group under (conventional) matrix multiplication.
This group is called the General Linear Group and is denoted $\operatorname{GL} \left({n, K}\right)$, or $\operatorname{GL} \left({n}\right)$ if the field is implicit.
The field itself is usually $\R$, $\Q$ or $\C$, but can be any field.
Proof
Taking the group axioms in turn:
G0: Closure
The matrix product of two $n \times n$ matrices is another $n \times n$ matrix.
The matrix product of two invertible matrices is another invertible matrix.
Thus $\operatorname{GL} \left({n, K}\right)$ is closed.
G1: Associativity
Matrix Multiplication is Associative.
G2: Identity
The Identity Matrix serves as the identity of $\operatorname{GL} \left({n, K}\right)$.
G3: Inverses
From the definition of invertible matrix, the inverse of any invertible matrix $\mathbf A$ is $\mathbf A^{-1}$.
$\blacksquare$
See also
Subgroups of the General Linear Group
- Special Linear Group
- Unitary Group
- Special Unitary Group
- Orthogonal Group
- Symplectic Group
- Triangular Matrix Groups
Related Groups
Sources
- Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 34 \ (2)$
- John F. Humphreys: A Course in Group Theory (1996): $\S 1$: Example $1.7$
