Question

The group of all non-singular \(n \times n\) matrices is known as the general linear group \(G L(n)\) and that with only real elements as \(G L(n, \mathbf{R}) .\) If \(\mathbf{R}^{*}\) denotes the multiplicative group of non-zero real numbers, prove that the mapping \(\Phi\) : \(G L(n, \mathbf{R}) \rightarrow \mathbf{R}^{*}\), defined by \(\Phi(\mathrm{M})=\operatorname{det} \mathrm{M}\), is a homomorphism. Show that the kernel \(\mathcal{K}\) of \(\Phi\) is a subgroup of \(G L(n, \mathbf{R})\). Determine its cosets and show that they themselves form a group.

Short answer

The mapping \(\Phi(M) = \operatorname{det}(M)\) is a homomorphism. The kernel \(\mathcal{K}\) of \(\Phi\) is a subgroup of \(GL(n, \mathbf{R})\). Its cosets form a group.

Step-by-step solution

Define the Mapping

Consider the mapping \(\Phi: GL(n, \mathbf{R}) \rightarrow \mathbf{R}^*\) defined by \(\Phi(M) = \operatorname{det}(M)\). We'll show that this mapping is a homomorphism.

Show Homomorphism Property

To show that \(\Phi\) is a homomorphism, we need to verify that \(\Phi(AB) = \Phi(A) \Phi(B)\) for any \(A, B \in GL(n, \mathbf{R})\). Note that \(\Phi(AB) = \operatorname{det}(AB)\). Using the property of determinants, \(\operatorname{det}(AB) = \operatorname{det}(A) \times \operatorname{det}(B)\). Therefore, \(\Phi(AB) = \operatorname{det}(A) \times \operatorname{det}(B) = \Phi(A) \Phi(B)\). This confirms that \(\Phi\) is a homomorphism.

Define the Kernel

The kernel \(\mathcal{K}\) of \(\Phi\) is defined as the set of all matrices \(M \) in \(GL(n, \mathbf{R})\) such that \(\Phi(M) = 1\). So, \(\mathcal{K} = \{ M \in GL(n, \mathbf{R}) \mid \operatorname{det}(M) = 1 \}\), which is the set of all matrices with determinant 1.

Verify Subgroup Property

To verify that \(\mathcal{K}\) is a subgroup, we need to check the subgroup criteria: closure, identity, and inverse. 1. **Closure**: If \(A, B \in \mathcal{K}\), then \(\operatorname{det}(A) = 1\) and \(\operatorname{det}(B) = 1\). Therefore, \(\operatorname{det}(AB) = \operatorname{det}(A) \operatorname{det}(B) = 1 \). Thus, \(AB \in \mathcal{K}\). 2. **Identity**: The identity matrix \(I\) in \(GL(n, \mathbf{R})\) has \(\operatorname{det}(I) = 1\), so \(I \in \mathcal{K}\). 3. **Inverse**: If \(A \in \mathcal{K}\), then \(\operatorname{det}(A) = 1\). The determinant of the inverse is \(\operatorname{det}(A^{-1}) = \operatorname{det}(A)^{-1} = 1\), hence the inverse \(A^{-1} \in \mathcal{K}\). Therefore, \(\mathcal{K}\) is a subgroup.

Determine Cosets

The cosets of \(\mathcal{K}\) in \(GL(n, \mathbf{R})\) are of the form \(A \mathcal{K}\) for \(A \in GL(n, \mathbf{R})\). Each coset corresponds to a non-zero real number \(\Phi(A)\), since \(\Phi(A) = \operatorname{det}(A)\) and \(\Phi(K) = 1\) for all \(K \in \mathcal{K}\). Thus, the cosets can be regarded as \(\{A \mathcal{K} \mid \operatorname{det}(A) = c \}, \) with \(c \in \mathbf{R}^*\).

Show Cosets Form a Group

To show that the cosets form a group, consider the set of all cosets \(\{A \mathcal{K} \mid \operatorname{det}(A) = c \}\). We define multiplication of cosets: \((A \mathcal{K})(B \mathcal{K}) = AB \mathcal{K}\). For any cosets \(A \mathcal{K}\) and \(B \mathcal{K}\), \(\operatorname{det}(AB) = \operatorname{det}(A) \operatorname{det}(B) = (c \times d)\), which is again a non-zero real number. This set with coset multiplication satisfies the group axioms (closure, identity, inverses, associativity) and forms a group under this operation.

Key concepts

Nonsingular Matrices

A matrix is called 'nonsingular' or 'invertible' if it has a well-defined inverse. Mathematically, an \( n \times n \) matrix \( M \) is nonsingular if there exists another \( n \times n \) matrix \( M^{-1} \) such that \( M \times M^{-1} = I \), where \( I \) is the identity matrix. For a matrix, being nonsingular implies that it has a non-zero determinant. This property is crucial because it ensures the matrix can be used in various matrix operations, such as solving systems of linear equations. Nonsingular matrices are the backbone of many advanced topics in linear algebra and group theory.

Matrix Determinant

The determinant of a matrix is a scalar value that provides important information about the matrix. For an \( n \times n \) matrix \( M \), the determinant, denoted as \( \operatorname{det}(M) \), represents a function that maps the matrix to a real number. The determinant can be calculated using various methods, such as cofactor expansion or row reduction. Some key properties of the determinant include:

• \( \operatorname{det}(A B) = \operatorname{det}(A) \operatorname{det}(B) \)\
• If \( \operatorname{det}(M) = 0 \), the matrix \( M \) is singular (non-invertible).
• \( \operatorname{det}(M^{-1}) = \operatorname{det}(M)^{-1} \) if \( M \) is nonsingular.

Determinants play a crucial role in defining matrix homomorphisms and understanding the properties of matrix groups.

Group Theory

Group theory is a branch of mathematics that studies the algebraic structures known as groups. A group is a set equipped with a binary operation satisfying four properties: closure, associativity, identity, and inverses. For example, in the context of linear algebra, the set of all nonsingular \( n \times n \) matrices forms a group under matrix multiplication. This group is known as the General Linear Group, denoted \( GL(n, \mathbb{R}) \). Understanding group theory helps in analyzing the algebraic structures that arise in various mathematical and physical contexts. Homomorphisms, which preserve the group operation, are critical tools in group theory.

Kernel

The kernel of a group homomorphism is the set of elements in the domain that map to the identity element of the codomain. For the homomorphism \( \Phi: GL(n, \mathbf{R}) \rightarrow \mathbf{R}^* \) defined by \( \Phi(M) = \operatorname{det}(M) \), the kernel \( \mathcal{K} \) consists of all matrices with determinant 1:

• \( \mathcal{K} = \{ M \in GL(n, \mathbf{R}) \mid \operatorname{det}(M) = 1 \} \)

This set \( \mathcal{K} \) is a subgroup of \( GL(n, \mathbf{R}) \) because it satisfies the necessary conditions: closure under multiplication, existence of identity, and existence of inverses. The kernel provides insight into the structure and properties of the homomorphism.

Cosets

In group theory, cosets are a way of partitioning a group into disjoint subsets. Given a subgroup \( H \) of a group \( G \), a left coset of \( H \) in \( G \) is a subset of the form \( gH = \{ gh \mid h \in H \} \) for some \( g \in G \). For the homomorphism \( \Phi: GL(n, \mathbf{R}) \rightarrow \mathbf{R}^* \), the cosets of \( \mathcal{K} \) inside \( GL(n, \mathbf{R}) \) can be written as:

• \( A \mathcal{K} \) where \( \operatorname{det}(A) = c \in \mathbf{R}^* \)

These cosets divide the general linear group based on the value of the determinant. The set of cosets \( \{ A \mathcal{K} \mid \operatorname{det}(A) = c \} \) forms a group under coset multiplication: \( (A \mathcal{K})(B \mathcal{K}) = AB \mathcal{K} \), illustrating how group structure is preserved in the cosets.