Let the linear transformation \(T: \mathbb{R}^3 \rightarrow \mathbb{R}^3\) be defined by \(T(x, y, z) = (2x, 3y, -z)\). Find the pre-image of the vector \((6, 9, -3)\) under this transformation.

Step 1 ：To find the pre-image of a vector under a linear transformation, we set the image equal to the transformation and solve for the original vector. So, we have \((2x, 3y, -z) = (6, 9, -3)\).

Step 2 ：This gives us the system of equations: \[\begin{cases} 2x = 6 \\ 3y = 9 \\ -z = -3 \end{cases}\]

Step 3 ：Solving this system of equations, we get \(x = 3\), \(y = 3\), and \(z = 3\)