Final answer:
The statement that the set of solutions of a homogeneous linear differential equation is the kernel of a linear transformation is true. Homogeneous linear differential equations form a vector space of functions that satisfy the equation, which is known as the kernel of the associated linear operator.
Step-by-step explanation:
True, the set of all solutions of a homogeneous linear differential equation is indeed the kernel of a linear transformation. A homogeneous linear differential equation has the form L(y) = 0, where L is a linear differential operator. When we talk about solutions to this equation, we are referring to the functions y that satisfy the condition L(y) = 0. The collection of all such functions y forms a vector space, known as the kernel or null space of L. A linear transformation, in this context, is the application of the differential operator L to the functions in the function space considered.
For example, consider the simple homogeneous linear differential equation d^2y/dx^2 - 4y = 0. The solutions to this equation form a two-dimensional vector space spanned by functions e^{2x} and e^{-2x}. These functions are in the kernel of the linear transformation defined by the differential operator d^2/dx^2 - 4.