Let’s consider a driftless control system. We define the Linear Map $L:C([t_{0},t_{1}],\mathbb{R}^{m})\to \mathbb{R}^{n}$ to be $$L(u)=\int\limits _{t_{0}}^{t_{1}}G(\tau)u(\tau) \, d\tau$$and we define $W:\mathbb{R}\times \mathbb{R}\to M_{n}(\mathbb{R})$ as: $$W(t_{0},t_{1})=\int\limits _{t_{0}}^{t_{1}}G(\tau)G^{T}(\tau) \, d\tau$$Then, we have that $$\text{Image}(L)=\text{Image}(W):=\{ W\eta: \eta\in\mathbb{R}^{n} \}$$