Transfer Function of a Linear System
The transfer function of a linear system is a mathematical representation that relates the output of the system to its input. It is an essential tool in analyzing and designing control systems, as it provides insight into the system's behavior and allows for the prediction of its response to various inputs.

Definition
The transfer function, denoted as H(s), is defined as the ratio of the Laplace transform of the output, Y(s), to the Laplace transform of the input, X(s), with all initial conditions zero. It is expressed as:

H(s) = Y(s) / X(s)

Where s is the complex frequency variable.

Explanation of Options
Let's now go through each option and determine whether it represents the correct definition of the transfer function.

a) Ratio of the output, vo (t), and input, vi (t):
This option does not fully capture the definition of the transfer function. The transfer function is not just a simple ratio of the output and input signals at a given time t, but rather a mathematical representation of their relationship in the Laplace domain.

b) Ratio of the derivatives of the output and the input:
This option also does not accurately describe the transfer function. While derivatives may be involved in the analysis of systems, the transfer function specifically relates the Laplace transforms of the output and input, not their derivatives.

c) Ratio of the Laplace transform of the output and that of the input with all initial conditions zero:
This option correctly represents the definition of the transfer function. It emphasizes the importance of Laplace transforms and specifies that the initial conditions of the system should be zero. This definition aligns with the mathematical representation of the transfer function as the ratio of Y(s) to X(s).

d) None of these:
This option is incorrect since option c accurately describes the transfer function.

Conclusion
The correct answer is option c. The transfer function of a linear system is the ratio of the Laplace transform of the output and that of the input with all initial conditions zero. This definition allows for the analysis and design of control systems by providing a mathematical representation of the system's behavior.