Question

Show that the set \(G=G L(n, \mathbb{R}) \times \mathbb{R}^{n}\) equipped with the "product" $$ (\mathrm{A}, \mathbf{u})(\mathrm{B}, \mathbf{v}) \equiv(\mathrm{AB}, \mathrm{A} \mathbf{v}+\mathbf{u}) $$ forms a group. This is called the affine group.

Short answer

The set \(G\) with elements \((A, u)\) where \(A\) is an invertible matrix and \(u\) is a vector, forms a group under the operation \((A, u)(B, v) \equiv(AB, Av+u)\). This is because the set is closed under the operation, the operation is associative, the set contains an identity element \((I, 0)\), and every element has an inverse \((A^{-1}, -A^{-1}u)\).

Step-by-step solution

Closure

Take any two elements from \(G\), say \((A, u)\) and \((B, v)\). We need to check that their 'product' is also an element of \(G\). Using the given rule, the 'product' of \((A, u)\) and \((B, v)\) is \((AB, Av+u)\). Since the product of two invertible matrices is an invertible matrix and the sum of two vectors is a vector, \((AB, Av+u)\) is indeed an element of \(G\). Therefore, \(G\) is closed under the operation.

Associativity

Take three elements from \(G\), say \((A, u)\), \((B, v)\), and \((C, w)\). We need to check that \((A, u)((B, v)(C, w)) = ((A, u)(B, v))(C, w)\). Using the operation rule, we see that both sides reduce to \((ABC, ABw+Av+u)\), proving associativity.

Identity

We need an element in \(G\) that, when 'multiplied' with any element in \(G\), gives the same element back. Check that \((I, 0)\), where \(I\) is the identity matrix and \(0\) is the zero vector, serves as the identity. Indeed, for any \((A, u) \in G\), we have \((I, 0)(A, u) = (IA, Iv+0) = (A, u)\) and \((A, u)(I, 0) = (AI, Au+0) = (A, u)\). So, \((I, 0)\) is the identity.

Inverse

For any \((A, u) \in G\), an inverse element needs to be found that, when 'multiplied' with \((A, u)\), gives the identity element \((I, 0)\). One can verify that \((A^{-1}, -A^{-1}u)\) is the inverse of \((A, u)\), since \((A, u)(A^{-1}, -A^{-1}u) = (AA^{-1}, A(-A^{-1}u)+u) = (I, 0)\) and \((A^{-1}, -A^{-1}u)(A, u) = (A^{-1}A, A^{-1}u - A^{-1}u) = (I, 0)\).

Key concepts

Group Theory

When we study algebra, group theory is a foundational concept that we cannot overlook. A group is a mathematical structure that consists of a set of elements along with an operation that combines any two of its elements to form another element. However, not just any set with a binary operation is a group. There are specific properties that must be met: closure, associativity, identity, and invertibility.

To be more precise, closure means that when you ‘combine’ two elements of the group using the group operation, the result is also within the group. For instance, if you're adding numbers, adding two numbers from your set will always give you another number in the same set. Associativity ensures that when we group the elements differently, we get the same result. Think about it like a puzzle; no matter how you arrange the pieces (as long as it is in the correct order), you get the same image at the end. The identity is like a neutral element in the group; it doesn't change anything when combined with another element. Lastly, for every element, there must be another that reverses its effect when combined, which we call the inverse.

In the exercise provided, we apply these principles to the affine group, demonstrating these group properties step by step, ensuring students understand how to verify the structure of a group.

Invertible Matrices

Matrices play a vital role in various fields of mathematics, including algebra, calculus, and statistics. An invertible matrix, also called a non-singular or nondegenerate matrix, is particularly special. To be considered invertible, a matrix must have an inverse, a unique matrix that, when multiplied with the original, yields the identity matrix.

Why is this concept important? Because when a matrix is invertible, it means that the linear system of equations it represents has a unique solution. In essence, invertible matrices can be thought of as mathematical functions that can be undone or reversed - much like having a word in a language that has an exact antonym.

The affine group exercise exemplifies the significance of invertible matrices in the context of group operations. We can see that when invertible matrices are combined with vector addition as in the problem statement, their products remain within the set of invertible matrices, satisfying the group property of closure.

Vector Space

A vector space is another fundamental concept in mathematics, especially in linear algebra. The term may prompt images of arrows in space, which is not far off. Technically, a vector space is a collection of vectors that can be scaled and added together following certain rules. These vectors can represent anything with magnitude and direction, such as physical forces or velocities, or even more abstract concepts.

The rules or properties that vectors must satisfy within a vector space include commutativity, associativity of addition, compatibility of scalar multiplication with field multiplication, and existence of an additive identity and additive inverses. In the context of the affine group problem, we are working with real vector spaces where vectors are combined with matrices to form a group. The concept of vector spaces is essential for understanding how the set, when paired with the right operation, can maintain the group structure as demonstrated in the step-by-step solution of the affine group exercise.