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If T: R³ R³ is a linear transformation such that then T = -3 12 -2 *(Q)-E}··(O)-E}··(O)-E] T = 3 T

If T: R³ R³ is a linear transformation such that then T = -3 12 -2 *(Q)-E}··(O)-E}··(O)-E] T = 3 T

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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If \( T : \mathbb{R}^3 \rightarrow \mathbb{R}^3 \) is a linear transformation such that \[ T \left( \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \right) = \begin{bmatrix} 3 \\ -4 \\ -1 \end{bmatrix}, \quad T \left( \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} \right) = \begin{bmatrix} -1 \\ 1 \\ 1 \end{bmatrix}, \quad T \left( \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \right) = \begin{bmatrix} -1 \\ 3 \\ 2 \end{bmatrix}, \] then \[ T \left( \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} \right) = \begin{bmatrix} -3 \\ 12 \\ -2 \end{bmatrix}. \]
Transcribed Image Text:If \( T : \mathbb{R}^3 \rightarrow \mathbb{R}^3 \) is a linear transformation such that \[ T \left( \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \right) = \begin{bmatrix} 3 \\ -4 \\ -1 \end{bmatrix}, \quad T \left( \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} \right) = \begin{bmatrix} -1 \\ 1 \\ 1 \end{bmatrix}, \quad T \left( \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \right) = \begin{bmatrix} -1 \\ 3 \\ 2 \end{bmatrix}, \] then \[ T \left( \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} \right) = \begin{bmatrix} -3 \\ 12 \\ -2 \end{bmatrix}. \]
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