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 vectors. T(X1,X2.X3.X4) = (×1 + 8×2, 0, 6x2 + X4, X2 – X4) ..... A = (Type an integer or decimal for each matrix element.)
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 vectors. T(X1,X2.X3.X4) = (×1 + 8×2, 0, 6x2 + X4, X2 – X4) ..... A = (Type an integer or decimal for each matrix element.)
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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Question
![**Title: Linear Transformation and Matrix Mapping**
**Objective:**
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 vectors.
**Problem Statement:**
Given the transformation:
\[
T(x_1, x_2, x_3, x_4) = (x_1 + 8x_2, 0, 6x_2 + x_4, x_2 - x_4)
\]
**Task:**
Find the matrix \( A \) such that the transformation can be represented in matrix form:
\[
A = \begin{bmatrix} \_ & \_ & \_ & \_ \\ \_ & \_ & \_ & \_ \\ \_ & \_ & \_ & \_ \\ \_ & \_ & \_ & \_ \end{bmatrix}
\]
*(Type an integer or decimal for each matrix element.)*
**Explanation:**
We need to determine the components of matrix \( A \) that will map a vector \( \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix} \) to \( T(x_1, x_2, x_3, x_4) \). To do this, each output component of \( T \) should match the result from multiplying \( A \) with the vector.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3d31c5b4-c6dc-4739-be04-6ff777f4b60f%2Fb662629c-a2ce-476d-9781-8f2dfa117821%2Fhqs28rr_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Linear Transformation and Matrix Mapping**
**Objective:**
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 vectors.
**Problem Statement:**
Given the transformation:
\[
T(x_1, x_2, x_3, x_4) = (x_1 + 8x_2, 0, 6x_2 + x_4, x_2 - x_4)
\]
**Task:**
Find the matrix \( A \) such that the transformation can be represented in matrix form:
\[
A = \begin{bmatrix} \_ & \_ & \_ & \_ \\ \_ & \_ & \_ & \_ \\ \_ & \_ & \_ & \_ \\ \_ & \_ & \_ & \_ \end{bmatrix}
\]
*(Type an integer or decimal for each matrix element.)*
**Explanation:**
We need to determine the components of matrix \( A \) that will map a vector \( \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix} \) to \( T(x_1, x_2, x_3, x_4) \). To do this, each output component of \( T \) should match the result from multiplying \( A \) with the vector.
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