How to show that two given functions are linearly dependent

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Let $f_1,\ldots,f_n$ be functions defined on a given interval $I$ and $n-1$ times differentiable. Let $W(f_1,\ldots,f_n)$ be the Wronskian of these functions. It is well known that:

if $f_1,\ldots,f_n$ are linearly dependent on $I$, then $W(f_1,\ldots,f_n)$ vanishes on $I$.

In general, the converse is not true: the vanishing of the Wronskian on $I$ does not imply that the functions are linearly dependent. However, if $f_1,\ldots,f_n$ are analytic, then these functions are linearly dependent on $I$ if and only if $W(f_1,\ldots,f_n)\equiv 0$ on $I$.

Since $f(x)=17$, $g(x)=\cos^2(x)$ and $h(x)=\cos(2x)$ are analytic on $\mathbb{R}$, we can check if they are linearly dependent by testing their Wronskian:

sage: f(x) = 17 
sage: g(x) = cos(x)^2 
sage: h(x) = cos(2*x) 
sage: funs = [f(x),g(x),h(x)] 
sage: bool(wronskian(*funs,x)==0)                                               
True

Thus, $f$, $g$ and $h$ are linearly dependent.

First an answer for the special case, then a more general answer.

Trigonometric functions

In the special case of the example in the question, this helps:

sage: f(x) = 17
sage: g(x) = cos(x)^2
sage: h(x) = cos(2*x)

sage: h(x).simplify_trig()
2*cos(x)^2 - 1

Linear independence of functions

Returning to the general question, one way to check for possible linear dependence relations, or for proofs of independence, would be to project to a finite-dimensional subspace and check there.

Form a vectors from the value of each function at a chosen collection of evaluation points.

If these vectors are linearly independent, then the functions are.

If there is a linear relation between these vectors, then it's worth checking if the linear relation in fact holds for the functions themselves.

Let us illustrate with the three functions above.

Choose a collection of evaluation points:

sage: xx = (0, pi/6, pi/4, pi/3, pi/2)

Form a vector for each function:

sage: u = vector([f(x) for x in xx])
sage: v = vector([g(x) for x in xx])
sage: w = vector([h(x) for x in xx])

To look for dependence relations, compute the left kernel of the matrix with these vectors as rows.

sage: m = matrix([u, v, w])
sage: K = m.kernel()
sage: K
Vector space of degree 3 and dimension 1 over Algebraic Real Field
Basis matrix:
[  1 -34  17]

This suggests that maybe f - 34*g + 17*h is zero.

Extract the coefficients:

sage: a, b, c = K.basis()[0]
sage: a, b, c
(1, -34, 17)

Check if the linear relation in fact holds for the functions:

sage: bool(a*f(x) + b*g(x) + c*h(x) == 0)
True