general linear group GL(n)

Take all the n x n matrices, then throw away every one that is not invertible. What survives is GL(n): the set of invertible matrices. It is the natural home for everything in this field, because a symmetry is something you can undo, and undoing is exactly having an inverse.

Precisely, GL(n, F) = { A in M_n(F) : det(A) != 0 }, for F = R or C. Under matrix multiplication this is a group: the product of invertibles is invertible (det multiplies), the identity matrix is the unit, and each A has the inverse A^-1. It is NOT a vector space — the zero matrix and most sums are not invertible — so think of it as a multiplicative world, not an additive one.

Geometrically GL(n) is the group of ALL invertible linear maps of F^n: every reversible stretch, shear, rotation, and reflection, in any combination. Every other matrix group in this chapter (orthogonal, unitary, special linear, symplectic) is a SUBGROUP carved out of GL(n) by demanding it preserve some extra structure.

A caveat about its shape: det is a continuous function, so the condition det != 0 makes GL(n,R) an OPEN subset of the n^2-dimensional space of matrices, and it splits into two pieces (det > 0 and det < 0) that you cannot travel between continuously. GL(n,C), by contrast, is connected. This connectivity question is the first hint that these groups have interesting geometry — they are Lie groups.