Group Homomorphism from M2(R) to R

The given group homomorphism is φ : M2(R) → R defined by φ(A) = trc(A), where M2(R) denotes the set of 2x2 matrices with real entries, and trc(A) denotes the trace of the matrix A.

Definition of Group Homomorphism

A group homomorphism is a function between two groups that preserves the group operation. In this case, the groups are M2(R) and R.

Kernel of the Homomorphism

The kernel of a homomorphism is the set of elements in the domain that map to the identity element in the codomain. In this case, we need to find the set of matrices A in M2(R) such that φ(A) = 0.

Finding the Kernel

Let A = [[a, b], [c, d]] be a matrix in M2(R). Then the trace of A is trc(A) = a + d.

To find the kernel, we need to solve the equation φ(A) = trc(A) = 0.

a + d = 0

This equation implies that a = -d.

So, the kernel consists of matrices of the form A = [[a, b], [-a, d]], where a and d are real numbers.

Isomorphism and the Kernel

To determine which map the kernel is isomorphic to, we need to consider the properties of the kernel.

Kernel of φ is isomorphic to the additive group of real numbers (R, +)

The kernel of φ consists of matrices of the form A = [[a, b], [-a, d]], where a and d are real numbers. This set forms an additive group isomorphic to the group of real numbers (R, +) under addition.

Explanation

The isomorphism between the kernel of φ and the additive group of real numbers (R, +) can be established by considering the following:

1. Identity Element: The zero matrix [[0, 0], [0, 0]] is the identity element in the kernel, which corresponds to the additive identity in the group of real numbers (R, +), i.e., 0.

2. Closure: Addition of matrices in the kernel results in another matrix in the kernel, and addition of real numbers in (R, +) results in another real number.

3. Inverses: Every matrix in the kernel has an inverse within the kernel, where the additive inverse of [[a, b], [-a, d]] is [[-a, -b], [a, -d]]. Similarly, every real number in (R, +) has an additive inverse.

4. Associativity and Commutativity: Addition of matrices in the kernel and addition of real numbers in (R, +) satisfy associativity and commutativity.

Therefore, the kernel of φ is isomorphic to the additive group of real numbers (R, +).

Summary

The kernel of the group homomorphism φ : M2(R) → R, given by φ(A) = trc(A), is isomorphic to the additive group of real numbers (R, +). This is because the elements in the kernel, matrices of the form A =