For $A \in \R^{n \times n}$ and $q \in \R^n$, the linear complementarity problem $LCP(A, q)$ is to determine if there is $x \in \R^n$ such that $x \geq 0,$ $y = Ax+q \geq 0$ and $x^T y = 0$. Such an $x$ is called a solution of $LCP(A, q)$. $A$ is called an $R_0$-matrix if $LCP(A, 0)$ has zero as the only solution. In this article, the class of $R_0$-matrices is extended to include typically singular matrices, by requiring in addition that the solution $x$ above belongs to a subspace of $\R^n$. This idea is then extended to semidefinite linear complementarity problems, where a characterization is presented for the multiplicative transformation.