Linear Independence Using the Wronskian

Using the Wronskian method, determine if the functions $y_1 = 2t - 1$, $y_2 = t^2 + 5$, and $y_3 = 4t - 7$ are linearly independent.

In this problem, we explore the concept of linear independence through the use of the Wronskian. The Wronskian is a determinant associated with a set of functions, often used in the context of differential equations to identify if a set of solutions is linearly independent. If the Wronskian is non-zero for some value of the variable, the functions are linearly independent over that interval. On the other hand, if the Wronskian is zero everywhere, the functions may be dependent, though additional analysis may be required for conclusive results.

Understanding linear independence is crucial because it helps in forming a basis for solution spaces in differential equations and systems of equations. These concepts are key to understanding the broader subject of linear algebra and its applications in differential equations. By examining the problem through the lens of the Wronskian, you learn to connect the geometric interpretation of linear independence with the algebraic tools used to test it. This is particularly useful when dealing with solutions of second-order linear homogeneous differential equations, where linear independence of solutions ensures a complete solution space.