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Exercise 10.1.2 Consider the following functions T: R³ → R2. For each function T, show that T is a linear transformation. Do this by showing that T is a matrix transformation, i.e., find a matrix A such that T(v) = Av. (a) T (b) T (c) T Z X Z X y Z (d) T y 1 1 = 2y-3x+z 7x+2y+z 3x - 11y+2z 3x+2y+z x+2y+6z 2y-5x+z x+y+z

Exercise 10.1.2 Consider the following functions T: R³ → R2. For each function T, show that T is a linear transformation. Do this by showing that T is a matrix transformation, i.e., find a matrix A such that T(v) = Av. (a) T (b) T (c) T Z X Z X y Z (d) T y 1 1 = 2y-3x+z 7x+2y+z 3x - 11y+2z 3x+2y+z x+2y+6z 2y-5x+z x+y+z

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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can you please solve (c) & (d) ?

**Exercise 10.1.2** Consider the following functions \( T: \mathbb{R}^3 \rightarrow \mathbb{R}^2 \). For each function \( T \), show that \( T \) is a linear transformation. Do this by showing that \( T \) is a matrix transformation, i.e., find a matrix \( A \) such that \( T(\mathbf{v}) = A\mathbf{v} \). (a) \( T \left( \begin{bmatrix} x \\ y \\ z \end{bmatrix} \right) = \begin{bmatrix} x + 2y + 3z \\ 2y - 3x + z \end{bmatrix} \). (b) \( T \left( \begin{bmatrix} x \\ y \\ z \end{bmatrix} \right) = \begin{bmatrix} 7x + 2y + z \\ 3x - 11y + 2z \end{bmatrix} \). (c) \( T \left( \begin{bmatrix} x \\ y \\ z \end{bmatrix} \right) = \begin{bmatrix} 3x + 2y + z \\ x + 2y + 6z \end{bmatrix} \). (d) \( T \left( \begin{bmatrix} x \\ y \\ z \end{bmatrix} \right) = \begin{bmatrix} 2y - 5x + z \\ x + y + z \end{bmatrix} \).
Transcribed Image Text:**Exercise 10.1.2** Consider the following functions \( T: \mathbb{R}^3 \rightarrow \mathbb{R}^2 \). For each function \( T \), show that \( T \) is a linear transformation. Do this by showing that \( T \) is a matrix transformation, i.e., find a matrix \( A \) such that \( T(\mathbf{v}) = A\mathbf{v} \). (a) \( T \left( \begin{bmatrix} x \\ y \\ z \end{bmatrix} \right) = \begin{bmatrix} x + 2y + 3z \\ 2y - 3x + z \end{bmatrix} \). (b) \( T \left( \begin{bmatrix} x \\ y \\ z \end{bmatrix} \right) = \begin{bmatrix} 7x + 2y + z \\ 3x - 11y + 2z \end{bmatrix} \). (c) \( T \left( \begin{bmatrix} x \\ y \\ z \end{bmatrix} \right) = \begin{bmatrix} 3x + 2y + z \\ x + 2y + 6z \end{bmatrix} \). (d) \( T \left( \begin{bmatrix} x \\ y \\ z \end{bmatrix} \right) = \begin{bmatrix} 2y - 5x + z \\ x + y + z \end{bmatrix} \).
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