Question

Let \(T: \mathbb{C} \rightarrow \mathbb{C}\) be a linear transformation of the real vector space \(\mathbb{C}\) and assume that \(T(a)=a\) for every real number \(a\). Show that the following are equivalent: a. \(T(z w)=T(z) T(w)\) for all \(z\) and \(w\) in \(\mathbb{C}\). b. Either \(T=1_{\mathbb{C}}\) or \(T(z)=\bar{z}\) for each \(z\) in \(\mathbb{C}\) (where \(\bar{z}\) denotes the conjugate).

Short answer

Both a and b are equivalent because both lead to T being either identity or conjugate.

Step-by-step solution

Understand the Problem

We are given a linear transformation \( T \) of the complex vector space \( \mathbb{C} \) that acts as the identity on real numbers. We need to show the equivalence of two statements: whether \( T \) is multiplicative on the entire complex space, and whether \( T \) is one of two possible functions (identity or conjugate).

Analyze Property a: Homomorphism for Multiplication

Assume that \( T(z w) = T(z) T(w) \) for all \( z, w \in \mathbb{C} \). This implies \( T \) behaves multiplicatively. So for \( z = x + yi \), consider \( T(1) = 1 \) and \( T(i^2) = iT(i) = -1 \). Thus \( T(i) \) must be \( i \) or \( -i \).

Determine Function Form with given Property a

Let \( T(z) = x + yi \), then \( T(z) = T(x + yi) = T(x) + yiT(i) = x + yiT(i) \). If \( T(i) = i \), then \( T(z) = z \); if \( T(i) = -i \), then \( T(z) = \bar{z} \). Thus, \( T \) is either the identity or the conjugate operator.

Analyze Property b: Specific Function Forms

Assume that either \( T(z) = z \) or \( T(z) = \bar{z} \) for each \( z \in \mathbb{C} \). These functions must naturally satisfy \( a \rightarrow T(z w) = T(z)T(w) \). For the identity, it is true as we naturally have \( z w = zw = z w \). Similarly, if \( T(z) = \bar{z} \), then \( T(z w) = \overline{z w} = \bar{z} \bar{w} = T(z)T(w) \). Thus proving the multiplicative behavior.

Conclusion on Equivalence

Both statements a and b are indeed equivalent under the given conditions. Statement a ensures that the transformation is multiplicative, and b shows that the functions fulfill this property naturally as they are inherent forms of linear transformations preserving this property.

Key concepts

Homomorphism

A homomorphism is an essential concept in abstract algebra, representing a structure-preserving map between two algebraic structures, such as groups, rings, or vector spaces. In our specific context, involving complex numbers, we are examining a homomorphism with respect to multiplication. When we say that a linear transformation \( T \) is a homomorphism for multiplication, it means the transformation respects the operation of multiplication among complex numbers.

In mathematical terms, for \( T(z w) = T(z) T(w) \) to hold for all complex numbers \( z \) and \( w \), \( T \) must maintain this multiplicative property consistently. This relationship means that \( T \) keeps the structure of multiplication intact, much like an identity or conjugate operation would. This structure-preserving characteristic is what qualifies \( T \) as a multiplicative homomorphism. Understanding this helps clarify why such transformations are key in preserving algebraic identities across different transformations.

Complex Numbers

Complex numbers extend the idea of one-dimensional number lines to a two-dimensional plane, introducing the concept of real and imaginary parts. Any complex number can be expressed as \( z = x + yi \), where \( x \) is the real component, and \( yi \) the imaginary part, with \( i \) representing the square root of \(-1\). Complex numbers are pivotal in many areas of mathematics and engineering, providing solutions to polynomial equations that lack real roots.

The significance of complex numbers in linear transformations is profound. They provide a multidimensional space to explore transformations that aren't confined to just real numbers. When we look at how transformations act on complex numbers, it underscores the richness and versatility of algebraic operations that can be defined in the complex plane. They are foundational in expressing solutions and exploring properties of mathematical systems beyond real number constraints.

Conjugation

Conjugation is a specific operation applied to complex numbers. For any complex number \( z = x + yi \), its conjugate is \( \bar{z} = x - yi \). Conjugation reflects the complex number across the real axis on the complex plane. It effectively changes the sign of the imaginary part while leaving the real part unchanged. This operation is particularly important because it retains certain properties, such as magnitude, involving the square root of the sum of the squares of the real and imaginary components.

In the context of linear transformations on complex numbers, conjugation serves as a distinct function form significant for preserving multiplicative structures. When a transformation \( T(z) \) takes the form of conjugation (i.e., \( T(z) = \bar{z} \)), it is inherently preserving multiplication through the property \( T(z w) = \overline{z w} = \bar{z} \bar{w} \), exemplifying a special case of homomorphism. This operation is crucial for illustrating how certain transformations can maintain specified algebraic structures while altering numerical configurations.