What is the meaning of "$X \mapsto AX$ is a surjective mapping from $\mathbb{R}^n$ to $\mathbb{R}^n$"? If square matrix $A$ has determinant $1$, then $X \mapsto AX$ is a surjective mapping from $\mathbb{R}^n$ to $\mathbb{R}^n$.

What does $X \mapsto AX$ mean?? 

Is it equivalent to say $T(x)=Ax$?
Because then $A$ would be invertible, and for a given $Y \in \mathbb{R}^n$, we have: $X = A^{-1}Y$, then $T(X) = T(A^{-1}Y) = AA^{-1}Y = I_nY = Y$, showing $T$ is surjective.
Because then $A$ would be invertible, and for a given $Y \in \mathbb{R}^n$, we have: $X = A^{-1}Y$, then $T(X) = T(A^{-1}Y) = AA^{-1}Y = I_nY = Y$, showing $T$ is surjective.