## Are Linear Transformations of Derivatives and Integrations Linearly Independent?

## Problem 463

Let $W=C^{\infty}(\R)$ be the vector space of all $C^{\infty}$ real-valued functions (smooth function, differentiable for all degrees of differentiation).

Let $V$ be the vector space of all linear transformations from $W$ to $W$.

The addition and the scalar multiplication of $V$ are given by those of linear transformations.

Let $T_1, T_2, T_3$ be the elements in $V$ defined by

\begin{align*}

T_1\left(\, f(x) \,\right)&=\frac{\mathrm{d}}{\mathrm{d}x}f(x)\\[6pt]
T_2\left(\, f(x) \,\right)&=\frac{\mathrm{d}^2}{\mathrm{d}x^2}f(x)\\[6pt]
T_3\left(\, f(x) \,\right)&=\int_{0}^x \! f(t)\,\mathrm{d}t.

\end{align*}

Then determine whether the set $\{T_1, T_2, T_3\}$ are linearly independent or linearly dependent.