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

[zhfs]

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?

 

[CompuChip]

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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