Problem 31 - Differential Equations and Linear Algebra

Question

Construct distinct matrix functions \(A\) and \(B\) defined on all of \(\mathbb{R}\) such that \(A(0)=B(0)\) and \(A(1)=B(1)\)

Step-by-Step Solution

Answer

The distinct matrix functions A and B can be represented as: \(A(x) = x \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} + \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\) \(B(x) = 2x \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} + \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\) These matrix functions satisfy the conditions \(A(0)=B(0)\) and \(A(1)=B(1)\), while still being distinct since they have different linear coefficients.

Step 1: Construct the Constant Matrix Functions

To create two distinct matrix functions that still have the same output values at the given specific points, we can select constant 2x2 matrices for \(A_0\) and \(B_0\) such that: \(A_0 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\) \(B_0 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\) This ensures that both matrix functions have the same value at 0, as: \(A(0) = a \times 0 + A_0 = A_0 = B_0 = b \times 0 + B_0 = B(0)\)

Step 2: Choose Linear Coefficients

Now, let us choose two linear coefficients, a and b, that will ensure the condition \(A(1)=B(1)\) is satisfied: a = 1 b = 2 Hence, the matrix functions A and B are: \(A(x) = x \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} + \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\) \(B(x) = 2x \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} + \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\) Clearly, A(x) and B(x) are distinct as they have different linear coefficients a and b, but they satisfy both conditions \(A(0)=B(0)\) and \(A(1)=B(1)\).

Key Concepts

Understanding Constant Matrix

A constant matrix in linear algebra is a matrix whose elements are all constants—that is, numbers that do not change with respect to any variables or parameters involved. For example, the identity matrix, which is used in the exercise above, is a constant matrix. It has ones on the diagonal and zeros elsewhere, like this:

\[ \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} \]
Constant matrices play several roles in matrix algebra, serving as a foundation or initial state from which more complex matrix functions can be constructed. Since their values don't change, they're often used as building blocks for functions like those we construct in the given exercise. Their predictability simplifies equations, making it easier to implement algebraic operations.

Linear Algebra Basics

Linear algebra is the branch of mathematics that deals with vectors, vector spaces, linear mappings, and systems of linear equations. It's essential in multiple areas, including engineering, physics, computer science, and economics. The matrices that we work with in linear algebra can represent linear transformations, and the operations like addition and multiplication follow certain algebraic rules.

The study of matrix functions, which you see in the provided exercise, is an extension of linear algebra that combines matrix theory with general function properties. This intertwines with concepts like linear independence, span, and basis, which are critical for understanding the structure and dimensions of vector spaces.

Constructing Matrix Functions

Matrix functions can be thought of as 'recipes' that transform input values into output matrices according to a specific rule or set of rules. In our case, constructing matrix functions to satisfy certain conditions necessitates a thoughtful choice of the components that make up the function. These components typically consist of variable parts and constant parts.

In the exercise, the construction of matrix functions A(x) and B(x) starts with selecting constant matrices that are equal at a certain input, x=0. This is followed by the addition of variable components—in this case, linear coefficients—that are designed to ensure another condition is met, that is, A(1)=B(1). The careful construction and balancing of these parts allow for the creation of distinct matrix functions that abide by the given requirements.

The Role of Linear Coefficients

In the context of our exercise, linear coefficients are the constants by which an input variable is multiplied. They play a crucial role in defining the linear aspect of a matrix function. In the simplest terms, they dictate how much the output will change when the input changes.

By choosing different linear coefficients for A(x) and B(x), we create two distinct functions that still manage to provide the same output at x=0 and x=1. This is an illustrative example of how linear coefficients determine the rate of change in a linear function. When constructing matrix functions, these coefficients can be strategically selected to fulfill specific conditions or to ensure that certain relationships between functions are preserved, all while keeping the matrix output within the function's range.