Problem 29 - Differential Equations and Linear Algebra

Question

Given an example of a matrix function of the specified form. (Many examples may be possible.) \(2 \times 1\) matrix function \(A\) that is nonconstant such that all elements of \(A(t)\) are in [0,1] for every \(t\) in \(\mathbb{R}\)

Step-by-Step Solution

Answer

A nonconstant \(2 \times 1\) matrix function A(t) with elements in the range \([0,1]\) can be given by: \(A(t) = \begin{bmatrix} \frac{1+\sin(t)}{2} \\ \frac{1+\cos(t)}{2} \end{bmatrix}\)

Step 1: Choose appropriate functions

To build a nonconstant 2x1 matrix function A(t) with all its elements within the range [0, 1], we will use the sine and cosine functions. First, consider the 2x1 matrix function: A(t) = \[\begin{bmatrix} \sin(t) \\ \cos(t) \end{bmatrix}\] Now, all elements of A(t) are within the range [-1, 1], and the next step is to adjust the range so that all the elements are within [0,1].

Step 2: Adjust the range

To adjust the range from [-1, 1] to [0, 1], we will add 1 to both the sine and cosine functions, and then divide by 2. This transformation will ensure that all elements of A(t) lie within the desired range. The result is the following 2x1 matrix function: A(t) = \[\begin{bmatrix} \frac{1+\sin(t)}{2} \\ \frac{1+\cos(t)}{2} \end{bmatrix}\] So, our example of a nonconstant 2x1 matrix function with the specified properties is: A(t) = \[\begin{bmatrix} \frac{1+\sin(t)}{2} \\ \frac{1+\cos(t)}{2} \end{bmatrix}\]

Key Concepts

Range Adjustment

Range adjustment is a common procedure in mathematical functions where you modify the output of a function to fit within a specific interval. This is particularly useful when dealing with functions whose natural output range does not meet the desired constraints.
In our matrix function example, both the sine and cosine functions naturally produce outputs ranging from -1 to 1. However, we need values that fall between 0 and 1.
To achieve this, a simple linear transformation is applied:

• Add 1 to shift the range from [-1, 1] to [0, 2].
• Divide by 2 to compress the range from [0, 2] to [0, 1].

This adjustment translates mathematically to:

• For sine: \ \( \frac{1 + \sin(t)}{2} \)
• For cosine: \ \( \frac{1 + \cos(t)}{2} \)

Place these adjusted terms in the matrix, obtaining elements that meet the range requirement of the problem.

Matrix Transformation

Matrix transformation involves modifying a matrix using mathematical operations to achieve a desirable property or form. It's an important concept in linear algebra, involving manipulations like scaling, translating, rotating, and reflecting matrices.
In our scenario, we are forming a 2x1 matrix function \( A(t) \) whose elements need to capture specific behaviors dictated by trigonometric functions.
The original matrix:\[ \begin{bmatrix} \sin(t) \ \cos(t) \end{bmatrix} \]captures the oscillatory nature of these trigonometric functions. However, we need to transform this matrix such that its elements adhere to a specific range constraint, in this case, [0, 1].

• This transformation is achieved by applying a series of operations to each element as illustrated in the range adjustment process.

These transformations are key in ensuring that the matrix meets specified requirements.

Trigonometric Functions

Trigonometric functions, like sine and cosine, are fundamental in mathematics, describing relationships in right-angled triangles and oscillatory phenomena such as waves.
They are periodic functions, meaning they repeat their values in regular intervals, specifically every \(2\pi\) for sine and cosine.
In our example, these functions are chosen because their properties allow easy manipulation into a desired range:

• The basic outputs vary cyclically between -1 and 1, which after transformation, can fit into any specified range.
• Their smooth periodic behavior makes them predictable and uniform across all real numbers \(t\).

By applying the range transformation, they are scaled and translated to ensure that every output is easily confined within our required parameters. Understanding these core properties makes them versatile tools in constructing matrix functions like \(A(t)\).