We prove that $L(X,Y)$ is complemented in $Lip_0(X, Y)$ (via a norm-one projection) provided that $Y$ is a dual space. Next, we introduce a vector-valued Lipschitz-free space $F_Y(X)$, a linear contraction $\beta_X^Y: F_Y(X) \to Y$ and prove that the quotient space ${ Lip_{ 0}(X, Y)}/{ L(X, Y) }$ is isometrically isomorphic to $L(\ker(\beta_X^Y), Y)$ whenever $Y$ is injective. We also consider a $Y$-valued duality pairing between $Lip_0(X, Y)$ and $F_Y(X)$ and obtain a necessary and sufficient condition for a Lipschitz map to be linear. As an application, we describe $\ker(\beta_{ \mathbb{ R}}^{ \mathbb{ R}})$ as the pre-dual of the quotient space $L^{ \infty}(\mathbb{ R})/\tilde{ \mathbb{ R}}$ where $\tilde{ \mathbb{ R}}$ is the set of all constant maps on $\mathbb{ R}$.
Categories: math.FA