Problem 5 Prove that the mapping \(g: \mat... [FREE SOLUTION]

Chapter 4: Problem 5

Prove that the mapping $g: \mathbb{C} \rightarrow \mathbb{C}, g(z)=-i z+1+2 i$ is an isometry. Analyze $g$ as in problem $4 .$

Short Answer

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Question: Prove that the function $g(z) = -i z + 1 + 2i$ is an isometry. _Answer: To prove that the function $g(z) = -i z + 1 + 2i$ is an isometry, we showed that for any complex numbers $z_1$ and $z_2$, the distance between them remains the same after applying the function, i.e., $|-i(z_1 - z_2)| = |z_1 - z_2|$. This confirms that the given function is indeed an isometry.

Step by step solution

Write down g(z) for z_1 and z_2

Given the function $g(z) = -iz + 1 + 2i$, let's find $g(z_1)$ and $g(z_2)$ for any two complex numbers $z_1$ and $z_2$: $$ g(z_1) = -i z_1 + 1 + 2i \\ g(z_2) = -i z_2 + 1 + 2i $$

Find g(z_1) - g(z_2)

Now let's find the difference between $g(z_1)$ and $g(z_2)$: $$ g(z_1) - g(z_2) = (-iz_1 + 1 + 2i) - (-iz_2 + 1 + 2i) \\ g(z_1) - g(z_2) = -i(z_1 - z_2) $$

Calculate the magnitudes of g(z_1) - g(z_2) and z_1 - z_2

Now we need to compare the magnitudes of $g(z_1) - g(z_2)$ and $z_1 - z_2$. So, we need to calculate $|-i(z_1 - z_2)|$ and $|z_1 - z_2|$ by using the definition of magnitude for a complex number: For $|-i(z_1 - z_2)| = |-i(x_1 - x_2 + i(y_1 - y_2))|$, where $z_1 = x_1 + iy_1$ and $z_2 = x_2 + iy_2$. Similarly, $|z_1 - z_2| = | x_1 - x_2 + i(y_1 - y_2)|$.

Prove |-i(z_1 - z_2)| = |z_1 - z_2|

To prove $|-i(z_1 - z_2)| = |z_1 - z_2|$, let's write down both expressions and compare them: $$ |-i(z_1 - z_2)| = |-i(x_1 - x_2 + i(y_1 - y_2))| = |-ix_1 + x_2 + y_1 - y_2 | = \sqrt{(-ix_1 + x_2)^2 + (y_1 - y_2)^2} \\ |z_1 - z_2| = |x_1 - x_2 + i(y_1 - y_2)| = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2} $$ We need to show that $|-i(z_1 - z_2)|^2 = |z_1 - z_2|^2$. In other words, we need to show that: $$ ((-ix_1 + x_2)^2 + (y_1 - y_2)^2) = (x_1 - x_2)^2 + (y_1 - y_2)^2 $$ Expanding, we get: $$ ((-1)^2(y_1 - y_2)^2 + 2i(ix_1 - x_2)(y_1 - y_2) + (x_1 - x_2)^2) = (x_1 - x_2)^2 + (y_1 - y_2)^2 $$ Simplifying, we can cancel out the equal terms on both sides: $$ ((-1)^2(y_1 - y_2)^2 + (x_1 - x_2)^2) = (x_1 - x_2)^2 + (y_1 - y_2)^2 $$ As we can see, the left-hand side and right-hand side are identical, which means that $|-i(z_1 - z_2)|^2 = |z_1 - z_2|^2$. Therefore, as desired $|-i(z_1 - z_2)| = |z_1 - z_2|$, proving that the mapping $g(z) = -i z + 1 + 2i$ is indeed an isometry.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Complex Numbers

Complex numbers are an extension of the real numbers and are fundamental in advanced mathematics and physics. A complex number is expressed in the form of a + bi, where a and b are real numbers, and i is the imaginary unit with the property that i² = -1.

In this context, a is the real part of the complex number, and b is the imaginary part. This two-part composition allows complex numbers to convey two dimensions of information, which is exceptionally useful in fields such as electrical engineering and fluid dynamics, where phenomena are often not adequately described by just real numbers.

Moreover, complex numbers can be represented as points or vectors in the two-dimensional plane commonly referred to as the complex plane. The horizontal axis in this plane is the real axis, while the vertical axis is the imaginary axis. This visual representation helps in understanding the behavior of complex numbers, especially when it comes to operations such as addition, multiplication, and transformations.

Magnitude of Complex Numbers

The magnitude (or modulus) of a complex number is a measure of its size and is denoted by |z|. For any complex number expressed as z = x + iy, its magnitude is given by |z| = $ \sqrt{x^2 + y^2} $, which essentially is the distance of the point (x, y) from the origin (0, 0) on the complex plane.

This distance is calculated using the Pythagorean theorem, which relates the legs of a right triangle to its hypotenuse. In our case, the legs are the real part x and the imaginary part y of the complex number, while the magnitude is the hypotenuse. Understanding the magnitude of complex numbers is crucial when comparing the sizes of different numbers or when working with complex number mappings, where preserving distance is often a key property.

Complex Number Mappings

Mappings or transformations involving complex numbers can be thought of as functions that take complex numbers as inputs and produce complex numbers as outputs. These functions can describe geometric transformations like rotations, translations, and reflections in the complex plane.

One special type of mapping is an isometry, which is a distance-preserving transformation. In other words, for an isometric mapping g(z), the distance between any two complex numbers z₁ and z₂ remains unchanged when those numbers are transformed by g. So, if |g(z₁) - g(z₂)| = |z₁ - z₂|, we say that g is an isometry.

For example, the mapping g(z) = -iz + 1 + 2i is an isometry because it preserves the magnitude of the difference between any two complex numbers. This means that no matter where you start in the complex plane, the distance between points after applying g is the same as the distance before applying g, which is a fascinating property used in various mathematical applications, especially in solving problems related to complex number theory and its applications in physics and engineering.

91影视

Prove that the mapping \(g: \mathbb{C} \rightarrow \mathbb{C}, g(z)=-i z+1+2 i\) is an isometry. Analyze \(g\) as in problem \(4 .\)

Short Answer

Step by step solution

Write down g(z) for z_1 and z_2

Find g(z_1) - g(z_2)

Calculate the magnitudes of g(z_1) - g(z_2) and z_1 - z_2

Prove |-i(z_1 - z_2)| = |z_1 - z_2|

Key Concepts

Understanding Complex Numbers

Magnitude of Complex Numbers

Complex Number Mappings

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