Question

If \(a\) and \(b\) are real numbers, define \(T_{a, b}: \mathbb{C} \rightarrow \mathbb{C}\) by \(T_{a, b}(r+s i)=r a+s b i\) for all \(r+s i\) in \(\mathbb{C}\) a. Show that \(T_{a, b}\) is linear and \(T_{a, b}(\bar{z})=\overline{T_{a, b}(z)}\) for all \(z\) in \(\mathbb{C}\). (Here \(\bar{z}\) denotes the conjugate of \(z\).) b. If \(T: \mathbb{C} \rightarrow \mathbb{C}\) is linear and \(T(\bar{z})=\overline{T(z)}\) for all \(z\) in \(\mathbb{C},\) show that \(T=T_{a, b}\) for some real \(a\) and \(b\).

Short answer

T_{a, b} is linear and satisfies T_{a, b}(\bar{z}) = \overline{T_{a, b}(z)}. Any such T is of form T_{a, b}.

Step-by-step solution

Show Linearity of T_{a,b}

To prove linearity of the transformation \(T_{a, b}\), we need to show that for all complex numbers \(z_1, z_2\) and a scalar \(c\), the function satisfies:\[T_{a, b}(z_1 + z_2) = T_{a, b}(z_1) + T_{a, b}(z_2)\]\[T_{a, b}(c \, z) = c \, T_{a, b}(z)\]**Step for Addition:**Let \(z_1 = r_1 + s_1 i\) and \(z_2 = r_2 + s_2 i\).\[T_{a, b}(z_1 + z_2) = T_{a, b}((r_1 + r_2) + (s_1 + s_2)i) = (r_1 + r_2)a + (s_1 + s_2)bi\]Applying the transformation separately:\[T_{a, b}(z_1) = r_1 a + s_1 bi\]\[T_{a, b}(z_2) = r_2 a + s_2 bi\]Thus:\[T_{a, b}(z_1) + T_{a, b}(z_2) = (r_1 a + r_2 a) + (s_1 b + s_2 b)i = (r_1 + r_2)a + (s_1 + s_2)bi\]**Step for Scalar Multiplication:**For a scalar \(c\) and a complex number \(z = r + si\):\[T_{a, b}(c \, z) = T_{a, b}((cr) + (cs)i) = (cr)a + (cs)bi\]\[c \, T_{a, b}(z) = c(r a + s bi) = (cr)a + (cs)bi\]Therefore, \(T_{a, b}\) is linear.

Prove Conjugate Property

We need to show that \(T_{a, b}(\bar{z}) = \overline{T_{a, b}(z)}\) for all complex numbers \(z\).Let \(z = r + si\). Then \(\bar{z} = r - si\):\[T_{a, b}(\bar{z}) = T_{a, b}(r - si) = ra - sbi\]\[T_{a, b}(z) = ra + sbi\]Taking the conjugate:\[\overline{T_{a, b}(z)} = \overline{ra + sbi} = ra - sbi\]Thus, \(T_{a, b}(\bar{z}) = \overline{T_{a, b}(z)}\).

Show Any Linear T with Conjugate Property is T_{a,b}

Given a linear transformation \(T\) such that \(T(\bar{z}) = \overline{T(z)}\), we need to show it is of the form \(T_{a, b}\).Since \(T\) is linear, we can express:\[T(r + si) = T(r) + T(si)\]For real numbers:\[T(r) = ra\] and \[T(si) = sbi\] where \(a\) and \(b\) are real.Hence:\[T(r + si) = ra + sbi\]Now, due to the conjugate property:\[T(\overline{(r + si)}) = T(r - si) = ra - sbi\]\[\overline{T(r + si)} = \overline{(ra + sbi)} = ra - sbi\]This confirms \(T(r + si) = ra + sbi\), confirming \(T = T_{a, b}\).

Key concepts

Complex Conjugate Properties

The complex conjugate of a complex number \( z = r + si \) is denoted as \( \bar{z} = r - si \). Understanding complex conjugates is essential because they have unique properties that are useful in complex analysis. One such property is that the conjugate of the sum of two complex numbers is equal to the sum of their conjugates:

• \( \overline{z_1 + z_2} = \overline{z_1} + \overline{z_2} \)

This means you can distribute the conjugation over addition. The conjugate of a product is similarly distributive:

• \( \overline{z_1 \cdot z_2} = \overline{z_1} \cdot \overline{z_2} \)

These properties aid in handling transformations involving complex numbers, such as linear transformation \( T_{a, b} \). For any transformation that maintains \( T(\bar{z}) = \overline{T(z)} \), it ensures that the form of the transformation preserves complex conjugate relationships. This is why verifying \( T_{a, b}(\bar{z}) = \overline{T_{a,b}(z)} \) is proof of the specific structure of transformation.

Linearity in Complex Functions

The concept of linearity is fundamental in mathematics, especially when dealing with functions from complex numbers to complex numbers. A transformation \( T_{a, b} \) is linear if, for all complex numbers \( z_1, z_2 \) and a scalar \( c \), it satisfies two conditions:

• Additivity: \( T_{a, b}(z_1 + z_2) = T_{a, b}(z_1) + T_{a, b}(z_2) \)
• Homogeneity: \( T_{a, b}(c \cdot z) = c \cdot T_{a, b}(z) \)

These properties imply that transformations like scaling and translating vectors in a linear fashion are consistent with our expectations from basic algebra. Additivity allows for distributing the transformation over an addition of inputs, simplifying complex operations into manageable steps. Homogeneity ensures that scaling of a complex number within the function translates to a predictable scaling of the output.Together, these properties characterize the function's form and behavior and are crucial when working with transformations that maintain these linear properties under conjugate manipulations.

Scalar Multiplication in Complex Functions

Scalar multiplication in complex functions is one of the two fundamental operations that define linear transformations. Given a complex number \( z = r + si \) and a real number \( c \), scalar multiplication within the transformation \( T_{a, b} \) operates as follows:

• When applied, \( T_{a, b}(c \cdot z) \) simplifies to \( (cr)a + (cs)bi \).

This means the scalar multiplies both the real and imaginary parts by the defined transformation coefficients \( a \) and \( b \), respectively.This operation ensures that any real number scales both parts of the complex number in a predictable manner, maintaining the structure of the original transformation. For example, if \( T_{a, b}(r + si) = ra + sbi \), then trivially, scaling \( z \) by \( c \) results in each part being scaled by \( c \), ensuring that the transformation remains linear. Understanding this symmetry and predictability is essential for solving problems involving complex linear mappings.