General linear group

The general linear group is a special collection of square grids called matrices. These grids have the same number of rows and columns, like a tiny square puzzle. These special matrices can be "flipped" or "reversed" in a special way, which mathematicians call being invertible.

These groups are important because they help mathematicians understand how things can be stretched, squished, or turned in space. They are used in many areas, from studying shapes and patterns to solving complex equations. The general linear group can be made with different types of numbers, like whole numbers, real numbers, or even more complex numbers.

One interesting part of these groups is a smaller group called the special linear group, where the special number in the corner of the matrix, called the determinant, is always 1. These groups help us understand symmetry and patterns in many different mathematical problems.

General linear group of a vector space

The general linear group of a vector space is a special group in mathematics. It includes all the ways you can stretch, shrink, or flip a space while keeping straight lines straight. These changes are called automorphisms.

When the space has a fixed size, called finite dimension, the general linear group can be shown using special arrangements of numbers called matrices. These matrices help us understand and work with these changes more easily.

In terms of determinants

A matrix can be turned around, or "inverted," if a special number called a determinant is not zero. This tells us which matrices can be used in the general linear group. When we work with certain number systems, we need to make sure the determinant can also be turned around within that system.

The general linear group includes all these special, invertible matrices, and they follow specific rules when multiplied together.

As a Lie group/algebra

The general linear group is about special square matrices that can move points in space in very organized ways. In simple terms, it’s a set of matrices that can be multiplied together and reversed, forming what mathematicians call a “group.”

This group is studied using a part of math called Lie groups, which looks at smooth shapes and how they change. For real numbers, this group has a certain size, and it includes matrices that can stretch or shrink space in regular ways. The group has interesting features, like how it can be broken into simpler parts, and it connects to other areas of geometry and algebra.

Special linear group

Main article: Special linear group

The special linear group is a special kind of group made from certain matrices. These matrices all have a determinant of 1, which means they follow a special rule. Because of this rule, they form a group — you can multiply them together and they still stay in the group.

These special matrices are important because they help describe changes in space that keep certain properties, like volume, the same. They are used in many areas of mathematics to study how things can be transformed while keeping some features unchanged.

Other subgroups

Diagonal matrices are special square matrices where the only numbers that are not zero are on the main diagonal. When these diagonal matrices can be changed back to regular matrices, they form a subgroup of the general linear group. In areas of math like the real numbers and complex numbers, these matrices can make space bigger or smaller in different directions.

Another important type is the scalar matrix. This is a diagonal matrix where every number on the diagonal is the same and not zero. These matrices form a subgroup that sits in the middle of the general linear group. They are important because they work well with all other matrices in the group.

Related groups and monoids

Projective linear group

Main article: Projective linear group

The projective linear group and the projective special linear group are made from some groups by taking away a few simple parts. They help us understand how certain shapes change when we move or twist them.

Affine group

Main article: Affine group

The affine group adds more moves to the general linear group. These moves shift every point in a space by the same distance. This helps us describe many usual ways to change shapes.

General semilinear group

Main article: General semilinear group

The general semilinear group includes changes that are almost straight but can also have twisting effects linked to the number system used. This group is useful when studying shapes and spaces.

Full linear monoid

If we take away one important rule from the general linear group, we get a new math idea called a monoid. This still works like the general linear group but allows extra kinds of changes.

Infinite general linear group

The infinite general linear group is a way to think about very large matrices that go on forever. It is made by linking smaller invertible matrices, with a 1 added in the bottom-right corner to connect them. This group is useful in a branch of math called algebraic K-theory, where it helps define something called K₁. When we use real numbers, this group has a clear structure thanks to a concept called Bott periodicity.

It is different from another type of group used for working with spaces called Hilbert spaces, which is larger and simpler in shape.