Show that T is a linear transformation by finding a matrix that implements the mapping. Note that x,, X2,... are not vectors but are entries in a vector. T(X1,X2 ,X3.X4) = 2×y + X2 - 2x3 - X4 (T: R4→R) ..... A =
Show that T is a linear transformation by finding a matrix that implements the mapping. Note that x,, X2,... are not vectors but are entries in a vector. T(X1,X2 ,X3.X4) = 2×y + X2 - 2x3 - X4 (T: R4→R) ..... A =
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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![**Show that \( T \) is a linear transformation by finding a matrix that implements the mapping. Note that \( x_1, x_2, \ldots \) are not vectors but are entries in a vector.**
\[ T \left( x_1, x_2, x_3, x_4 \right) = 2x_1 + x_2 - 2x_3 - x_4 \]
\[ (T: \mathbb{R}^4 \rightarrow \mathbb{R}) \]
----------------------
\[ A = \left[ \ \ \right] \]
**Explanation:**
The task is to demonstrate that the transformation \( T \), which is defined as a linear combination of the components \( x_1, x_2, x_3, x_4 \), can be represented using a matrix. Specifically, it involves determining a matrix \( A \) that corresponds with the linear transformation expressed in the given functional equation. The transformation \( T \) maps a vector from the four-dimensional real space \( \mathbb{R}^4 \) to the real numbers \( \mathbb{R} \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F704a64b5-5250-41d0-9c29-5aaf5a50e535%2Fd3243cbf-009f-4ca0-a632-f08fa715a707%2Fqspd7t_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Show that \( T \) is a linear transformation by finding a matrix that implements the mapping. Note that \( x_1, x_2, \ldots \) are not vectors but are entries in a vector.**
\[ T \left( x_1, x_2, x_3, x_4 \right) = 2x_1 + x_2 - 2x_3 - x_4 \]
\[ (T: \mathbb{R}^4 \rightarrow \mathbb{R}) \]
----------------------
\[ A = \left[ \ \ \right] \]
**Explanation:**
The task is to demonstrate that the transformation \( T \), which is defined as a linear combination of the components \( x_1, x_2, x_3, x_4 \), can be represented using a matrix. Specifically, it involves determining a matrix \( A \) that corresponds with the linear transformation expressed in the given functional equation. The transformation \( T \) maps a vector from the four-dimensional real space \( \mathbb{R}^4 \) to the real numbers \( \mathbb{R} \).
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