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I have to see if three linear operators A, B and C from the vector space of all linear operators from R^2 to R^3 are linearly independent. The mappings are all known (i.e. A(x, y) = ... , etc.).

Well, I set up an equation aA + bB + cC = 0, or, more precise, (aA + bB + cC)(x, y) = 0(x, y), where a, b and c are scalars. This equation should hold for all ordered pairs (x, y) from R^2 in order for these operators to be equal. Now, this equation led to a system of three equations with three unknowns, a, b and c (of course, the coefficients of the system are terms involving linear combinations of x and y). Now, since this must hold for every (x, y) from R^2, my logic was to plug in any (x, y) and solve for a, b and c. If the solution is trivial, then the operators are independent.

Nevertheless, there is obviously something wrong with my logic, since, after plugging in, for example (x, y) = (0, 1), I obtain a solution in parametric form, i.e. there is no unique trivial solution!

Any help is appreciated.