Real General Linear Group

The Real General Linear Group, denoted as $ GL(n, \mathbb{R}) $, encompasses all $ n \times n $ square matrices with real number entries that are invertible, meaning their determinant is not zero ($ \det(A) \neq 0 $). $$ GL(n,R) = \{ A \in M_n(R): \det(A) \ne 0 \} $$

Each matrix within $ GL(n, \mathbb{R}) $ signifies an invertible linear transformation within a real vector space $ \mathbb{R}^n $, essentially mapping the space onto itself.

These transformations play a pivotal role across a wide array of mathematical fields and their applications, from geometry to theoretical physics and engineering. They encompass operations like rotations, reflections, scaling, and more, all of which can be combined and reversed, demonstrating their versatility and fundamental importance.

In this context, the "group" concept is an algebraic structure meeting four specific criteria:

Closure: The combination of any two group elements (here, multiplying two matrices) yields another group member.
Associativity: The order in which elements are combined (or matrices are multiplied) has no impact on the outcome.
Identity Element: There’s an element in the group that, when combined with any other element, retains its original value. For matrices, this is the identity matrix $ I_n $, satisfying $ I_n \times A = A \times I_n = A $.
Invertibility: Each group member possesses an inverse that, when merged with its original, results in the identity element. This implies that for every matrix $ A $, there exists a matrix $ B $ such that $ A \times B = B \times A = I_n $.

The Real General Linear Group's structure embodies the full suite of possible invertible linear transformations, marking it as a cornerstone for study within linear algebra and group theory.

Example

Let's dive into a tangible example of the Real General Linear Group $ GL(n, \mathbb{R}) $ by focusing on $ GL(2, \mathbb{R}) $. This group consists of 2x2 matrices with real coefficients that have a non-zero determinant.

Consider the following matrix in $ GL(2, \mathbb{R}) $:

$$ A = \begin{pmatrix} 3 & 2 \\ 1 & 4 \end{pmatrix} $$

For a matrix to be part of $ GL(2, \mathbb{R}) $, its determinant must not be zero. In this case, the determinant is:

$$ \det(A) = 3 \times 4 - 2 \times 1 = 12 - 2 = 10 $$

With $ \det(A) \neq 0 $, matrix $ A $ is indeed invertible, securing its place in $ GL(2, \mathbb{R}) $.

The inverse of matrix A, noted as A^-1, can be calculated as follows:

$$ A^{-1} = \frac{1}{ \det{A} } \times \text{cof}(A)^T $$

$$ A^{-1} = \frac{1}{ 10 } \times \begin{pmatrix} 4 & -2 \\ -1 & 3 \end{pmatrix} $$

$$ A^{-1} = \begin{pmatrix} \frac{2}{5 } & -\frac{1}{5 } \\ -\frac{1}{10 } & \frac{3}{10 } \end{pmatrix} $$

Indeed, when matrix A is multiplied by its inverse A^-1, the identity matrix is the result:

$$ A \times A^{-1} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} $$

This matrix illustrates a linear transformation of the plane $ \mathbb{R}^2 $, offering a geometric interpretation.

The factor of 3 in the first column signifies scaling along the x-axis.
The factor of 4 in the second column signifies scaling along the y-axis.
The off-diagonal elements, 2 and 1, introduce a shear or rotation, shifting the standard basis vectors from their regular alignment along the axes.

To visualize this matrix's effect on the plane, consider multiplying any vector $ v = \begin{pmatrix} x \\ y \end{pmatrix} $ by $ A $:

$$ A \cdot v = \begin{pmatrix} 3 & 2 \\ 1 & 4 \end{pmatrix} \cdot \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 3x + 2y \\ x + 4y \end{pmatrix} $$

This process transforms the vector $ v $, scaling, rotating, reflecting, shearing, or applying a combination thereof, depending on $ x $ and $ y $'s specific values.

Indeed, $ GL(2, \mathbb{R}) $ encompasses all invertible linear transformations of the plane, illustrating the vast potential of linear transformations.