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Homework Statement
Given the system of vectors [itex]\cos x, \cos (x+2), \sin (x-5)[/itex]. Determine whether the system is linearly independent.
Homework Equations
The Attempt at a Solution
If it were linearly dependent, there would exist a non-trivial linear combination, such that:
[itex]k_1\cos x + k_2\cos (x+2) + k_3\sin (x-5) = 0[/itex]. Do not suggest the Wronski determinant, please - we haven't covered it in our Algebra, yet. There's got to be a simpler way to justify the system's (in)dependence.
If I fix any two constants as 0, then the third one will automatically be zero, as the equality has to be true for every [itex]x\in\mathbb{R}[/itex], therefore leaving us only with a trivial solution.
There is also a Lemma in the textbook that says whenever a sub-system turns out to be linearly dependent (provided the sub-system consists of at least two vectors, as according to another Lemma, the solution is automatically trivial if the sub-system consists of only one vector) then from there it follows the whole system is linearly dependent.
I'm quite sure the opposite isn't true - that if any sub-system is not linearly dependent then the whole system is linearly independent.
I'm not sure what to do here.