91Ó°ÊÓ

In the context of differential equations, a vector solution is a set of functions that together form a solution to a system of differential equations. This system is defined so that the derivative of a vector, with respect to a variable like time, matches the product of a matrix and the vector itself. To solve these types of systems, we look for vector functions that satisfy both the given equation and its specific structure. In our problem, the vector solution, \( \mathbf{X} \), is proposed as: \[ \mathbf{X} = \begin{pmatrix} 1 \ 3 \end{pmatrix} e^{t} + \begin{pmatrix} 4 \ -4 \end{pmatrix} t e^{t} \]- Each element in the vector \( \mathbf{X} \) is itself a function of \( t \). - The vector contains exponential components, which means it involves the exponential function \( e^t \).Identifying and verifying vector solutions requires one to ensure that the derivative of this vector, when appropriately manipulated, conforms to the original differential equation.

Matrix multiplication is a crucial operation in linear algebra, especially when dealing with systems of linear equations like the ones involving differential equations. Here, we deal with multiplying matrices by vectors. For our exercise, we have a matrix:\[ A = \begin{pmatrix} 2 & 1 \ -1 & 0 \end{pmatrix} \]We multiply this matrix by each component of the proposed solution vector separately and then combine the results.

• For \( \begin{pmatrix} 1 \ 3 \end{pmatrix} e^{t} \), calculate the product: \[ A \begin{pmatrix} 1 \ 3 \end{pmatrix} e^{t} = \begin{pmatrix} 5 \ -1 \end{pmatrix} e^{t} \]
• For \( \begin{pmatrix} 4 \ -4 \end{pmatrix} t e^{t} \), the product is: \[ A \begin{pmatrix} 4 \ -4 \end{pmatrix} t e^{t} = \begin{pmatrix} 4 \ -4 \end{pmatrix} t e^{t} \]

In matrix multiplication, each entry in the resulting vector or matrix is determined by specific dot-product calculations. This operation enables us to express the changes in each component of the vector solution as they are affected by the linear system defined by matrix \( A \).

Linear algebra provides the foundational framework for working with vector solutions and matrix multiplications. It involves working with subtle mathematical structures like matrices and vectors and understanding their relationships. At its core, linear algebra examines the solutions to equations that can be written in matrix form, such as \( A \mathbf{x} = \mathbf{b} \). Here, however, we are dealing with a differential system of the form:\[ \mathbf{X}' = A \mathbf{x} \] The solution of this form involves key concepts of linear independence, dimension, and transformations. - Vectors represent points in space or directions, while matrices represent transformations that can rotate, stretch, or compress those vectors.- Eigenvectors and eigenvalues, although not directly elaborated in this problem, play a big role in determining stable solutions to differential systems.The linear relationships maintained in these vector equations affirm the critical interconnectedness laid out by linear algebraic properties.

The verification process for vector solutions to differential equations involves ensuring that the derived solution satisfies the original equation. For our problem, we begin by differentiating the vector \( \mathbf{X} \) to find its derivative \( \mathbf{X}' \), and then perform matrix multiplication \( A \mathbf{x} \). The primary goal is to validate that both yield the same result.

• Start by computing the derivative of each element in \( \mathbf{X} \), resulting in:\[ \mathbf{X}' = \begin{pmatrix} (5+4t)e^{t} \ (-1-4t)e^{t} \end{pmatrix} \]
• Then, perform the matrix multiplication which results in:\[ A \mathbf{x} = \begin{pmatrix} (5+4t)e^{t} \ (-1-4t)e^{t} \end{pmatrix} \]

Upon matching \( \mathbf{X}' \) with \( A \mathbf{x} \), the verification confirms the proposed solution's validity. Thus, this meticulous process confirms that the solution derived is consistent with the given differential equation. This step-by-step check ensures no step in the solution process was overlooked or incorrect.