General linear group ($GL(V)$)

The general linear group $GL(V)$ is the group of invertible linear maps from a vector space $V$ to itself, with composition as its operation.

The unitriangular group ${\mathrm{UT}}_n(F)$ consists of upper triangular matrices over a field $F$ with every diagonal entry equal to one. Every such matrix is $I_n+N$ for a strictly upper triangular matrix $N$. Products have the same form, and $N^n=0$ gives the inverseThus these matrices form a subgroup of the general linear group. The case ${\mathrm{UT}}_3({\mathbb{F}}_3)$ is noncommutative but every element has cube equal to the identity matrix: $(I_3+N{)}^3=I_3$ because $N^3=0$ and $3=0$ in the finite field ${\mathbb{F}}_3$.

The real invertible matrices with positive determinant form a subgroup of the general linear group. The identity has determinant one, multiplicativity of the determinant preserves positivity under products, and $\det (A^{-1})=1/\det A>0$ preserves it under inversion. More precisely this is the kernel of a group homomorphism $A\mapsto \mathrm{sgn}(\det A)$ to $\{1,-1\}$, so it is a normal subgroup of index two for $n\ge 1$. Its elements preserve an orientation of a vector space.

A monomial matrix has exactly one nonzero entry in each row and column. It factors as an invertible diagonal matrix times a permutation matrix. Such matrices permute the coordinate axes and normalize the group of invertible diagonal matrices; over very small fields that normalizer can be larger if the diagonal subgroup fails to distinguish the axes.

The integral general linear group consists of integer matrices with determinant $\pm 1$. These are exactly the integer matrices whose inverses also have integer entries, so they are the automorphisms of the free abelian group ${\mathbb{Z}}^n$.

A matrix group is a group represented by invertible matrices, equivalently a subgroup of a general linear group.

A classical group is a standard family of linear groups, principally the general/special linear groups and groups preserving nondegenerate alternating, symmetric, Hermitian or quadratic forms. Finite-field versions yield many finite simple groups after imposing the appropriate determinant or derived-subgroup condition and quotienting the scalar center of a group. Small dimensions and small fields have exceptional isomorphisms and nonsimple cases.

The group of invertible linear maps preserving a specified quadratic form over a finite field. In characteristic two one must preserve the quadratic form itself, not merely its polar alternating bilinear form. For a nonsingular form, any two nonzero isotropic vectors can be incorporated into hyperbolic pairs and standard adapted bases, producing an isometry between them.