In this paper, for $n\ge 2$ and $n\ne 3$, the module algebra structures of $X_q(A_1)$ on the quantum $n$-space were discussed, where the quantum $n$-space is denoted by $A_q(n)$. In particular, a complete list of $X_q(A_1)$-module algebra structures on the quantum plane $A_q(2)$ was produced and the isomorphism classes of these structures were described.