One exercise on subspaces

Exercise 1 Prove that set of differentiable functions on {[a, b]} and satisfying {f' + 4f = 0} that is a subspace of {C_{[a, b]}}. Find a basis of this subspace and compute it’s dimension.

Hint: Let {W} be above set. Suppose that {f, g \in W}, we have {\begin{cases} f' + 4f&=0\\ g' + 4g&=0 \end{cases}}. Hence, {h = \alpha f + \beta g} also satisfies differential equation {h' + 4h = 0}. This implies {W} is a subspace of {C_{[a, b]}}.

{f' + 4f = 0 \Leftrightarrow \dfrac{df}{f} = -4dx \Leftrightarrow f(x) = C.e^{-4x}}. Each of vectors {f} is represented by the function {e^{-4x}, x \in [a, b]}, that implies the basis and the dimension of this subspace. \Box

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