Posted in Linear Algebra

# 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. $\Box$