Is the Set of Solutions to a Homogeneous Differential Equation a Vector Space?

AI Thread Summary
The discussion focuses on determining if the set of solutions to the homogeneous differential equation 7f''(x) + 4f'(x) - 6f(x) = 0 forms a vector space. Participants suggest using the subspace theorem to verify this by checking closure under addition and scalar multiplication. Key points include confirming that the zero function is a solution and that the sum of any two solutions, as well as any scalar multiple of a solution, remains a solution. The conversation emphasizes the importance of these properties in establishing the vector space criteria. Ultimately, the set of solutions is indeed a vector space if these conditions are satisfied.
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Homework Statement



use the subspace theorem to decide if the sets is a real vector space with respect to the usual operation

the set of all solutions of the homogenous differential equation
7f''(x) +4f'(x) -6f(x) = 0

Homework Equations



none

The Attempt at a Solution



try to put this second order d.e. into first order matrix system.
but don't know what to do next.

how to proof one matrix is under a vector space?
 
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You don't need to know the solutions. Just check if it's a vector space.
Is the function that sends every x to 0 a solution?
If f and g are solutions and r is any number, are (f + g) and r f solutions?
...
 

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