How to describe a three dimensional range. Consider the function $\mathbf{f}:\mathbf{R}^2\rightarrow\mathbf{R}^3$ given by $\mathbf{f}(\mathbf{x})=A\mathbf{x}$, where $A=\begin{bmatrix}
         2 & -1 \\
         5 &  0 \\
        -6 &  3
      \end{bmatrix}$ and the vector $\mathbf{x}$ in $\mathbf{R}^2$ is written as the $2\times1$ column matrix $\mathbf{x}=\begin{bmatrix}
        x_1 \\
        x_2
      \end{bmatrix}$. Describe the range of $\mathbf{f}$.
Clearly, $$
\mathbf{f}(\mathbf{x})=\begin{bmatrix}
               2 & -1 \\
               5 &  0 \\
              -6 &  3
            \end{bmatrix}\begin{bmatrix}
              x_1 \\
              x_2
            \end{bmatrix}=\begin{bmatrix}
               2x_1 - x_2  \\
               5x_1        \\
              -6x_1 + 3x_2
            \end{bmatrix}.
$$ However, I am uncertain as to how to "describe" the range of $\mathbf{f}$. Do I describe each component individually?

Hint : 
In such cases it is best to describe the range as the set {$(x,y,z) |$.. conditions on x,y and z}.  In this particular case, can x,y, and z take all the values freely or are there any constraints ? . When you reduce the equation to RREF. you will probably see one constraint. Express it in terms of a free variable.
