(a) Suppose 𝐮1(t), …, 𝐮n(t) are vector-valued functions...

(a) Suppose $\mathbf{u}_1(t), \ldots, \mathbf{u}_n(t)$ are vector-valued functions whose values at each point $t$ are linearly independent vectors in $\mathbb{R}^n$. Show that they form a basis for the solution space of a homogeneous constant coefficient linear system $\mathbf{u}=A \mathbf{u}$ if and only if each $d \mathbf{u}_j / d t$ is a linear combination of $\mathbf{u}_1(t), \ldots, \mathbf{u}_n(t)$. Hint: Use Exercise 10.4.28. (b) Show that a function $\mathbf{u}(t)$ belongs to the solution space of a homogeneous constant coefficient linear system $\dot{\mathbf{u}}=A \mathbf{u}$ if and only if $\frac{d^n \mathbf{u}}{d t^n}$ is a linear combination of $\mathbf{u}, \frac{d \mathbf{u}}{d t}, \ldots, \frac{d^{n-1} \mathbf{u}}{d t^{n-1}}$. Hint: Use Exercise 10.1.7.

Key Concepts

Homogeneous Linear Differential Systems

A homogeneous linear differential system with constant coefficients is an ordinary differential equation (or system of equations) in which the derivative(s) of an unknown function are expressed as a linear combination of the function itself, with constant coefficients. This characteristic allows the use of techniques such as eigenvalue analysis and matrix exponentials to obtain the general solution, and the structure of the system ensures that the superposition principle holds.

Solution Space and Its Basis

The solution space of a differential system is the set of all possible solutions that satisfy the homogeneous equation. A set of functions forms a basis for this space if they are linearly independent and every solution can be uniquely expressed as a linear combination of these basis elements. This concept is central to understanding the full set of behaviors permitted by the differential equation.

Constant Coefficient Systems

Constant coefficient systems are a subclass of linear differential equations where the coefficients in front of the unknown function and its derivatives do not change with respect to the independent variable. This property simplifies both the analysis and solution of the system, often allowing direct methods such as exponential ansatz or Laplace transforms to be applied effectively.

Linear Independence of Vector Functions

Linear independence for a set of vector-valued functions means that no function in the set can be written as a linear combination of the others for all values of the independent variable. This property is crucial for ensuring that a chosen set of solutions forms a valid basis for the solution space of the differential equation, enabling unique representation of any solution in terms of the basis.

Derivatives Expressed as Linear Combinations

The condition that derivatives of the basis functions can be expressed as linear combinations of the basis functions themselves is a manifestation of the invariance of the solution space under differentiation. This concept is particularly important in constant coefficient systems, where it implies that the space is closed under the operation of taking derivatives, a property that is used to verify that a set of functions truly forms a basis for the solution space.

Higher Order Derivative Characterization

In the context of linear differential equations of order n, a function can be shown to be a solution if its nth derivative can be represented as a linear combination of the function and its lower order derivatives. This idea underpins the method of constructing the minimal (or characteristic) polynomial that the function satisfies, and illustrates how the structure of the differential equation controls the recursive relationship among the derivatives.

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