Math  /  Algebra

QuestionSuppose that T:R3R3T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3} is such that its action on a vector [xyz]\left[\begin{array}{l}x \\ y \\ z\end{array}\right] is given below: T[xyz]=[xy2x3y2z2xy+z]T\left[\begin{array}{l} x \\ y \\ z \end{array}\right]=\left[\begin{array}{c} x-y \\ 2 x-3 y-2 z \\ 2 x-y+z \end{array}\right]
Find the inverse transformation T1T^{-1} and give its action on a general vector [xyz]\left[\begin{array}{l}x \\ y \\ z\end{array}\right]. T1[xyz]=[000]T^{-1}\left[\begin{array}{l} x \\ y \\ z \end{array}\right]=\left[\begin{array}{l} 0 \\ 0 \\ 0 \end{array}\right]

Studdy Solution
Express the action of T1 T^{-1} on a general vector:
T1[xyz]=111[122131021][xyz] T^{-1}\left[\begin{array}{l} x \\ y \\ z \end{array}\right] = \frac{1}{-11} \begin{bmatrix} 1 & 2 & 2 \\ -1 & -3 & -1 \\ 0 & -2 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix}
Perform the matrix multiplication to express the inverse transformation.
The inverse transformation T1 T^{-1} is:
T1[xyz]=[111(x+2y+2z)111(x3yz)111(2y+z)] T^{-1}\left[\begin{array}{l} x \\ y \\ z \end{array}\right] = \begin{bmatrix} \frac{1}{-11}(x + 2y + 2z) \\ \frac{1}{-11}(-x - 3y - z) \\ \frac{1}{-11}(-2y + z) \end{bmatrix}

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