Define a linearly independent set. Using any method you prefer, prove that the two functions 2x, x2are linearly independent.

A set of vectors (or functions, in this case) is said to be linearly independent if no vector in the set can be written as a linear combination of the other vectors. In other words, there are no redundant vectors in the set; each vector adds some new direction that cannot be reached by any combination of the other vectors.

To prove that the functions 2x and x^2 are linearly independent, we can use the Wronskian method. The Wronskian of a set of functions is a determinant used to test whether the functions are linearly dependent or independent.

For two functions f(x) and g(x), the Wronskian is defined as:

W(f,g) = f(x)g'(x) - f'(x)g(x)

Let's compute the Wronskian for our functions f(x) = 2x and g(x) = x^2.

First, we need to find the derivatives of f and g:

f'(x) = 2 g'(x) = 2x

Now we can compute the Wronskian:

W(f,g) = f(x)g'(x) - f'(x)g(x) = 2x * 2x - 2 * x^2 = 4x^2 - 2x^2 = 2x^2

The Wronskian is not identically zero, so the functions f(x) = 2x and g(x) = x^2 are linearly independent.