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.

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Accepted Answer

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)

Answer

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)

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.

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