21 + x2Let TDetermine if the specified linear transformation isx2 + x313 + x4(a) one-to-one. Justify your answer.(b) onto. Justify your answer.

Given the linear transformation Tx1x2x3x4=0x1+x2x2+x3x3+x4 Claim:…

Answered: -- Let T Determine if the specified…

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

21 + x2
Let T
Determine if the specified linear transformation is
x2 + x3
13 + x4
(a) one-to-one. Justify your answer.
(b) onto. Justify your answer.

Expert Solution

Part (a)

Given the linear transformation

$T ([[\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{matrix}]]) = [\begin{matrix} 0 \\ x_{1} + x_{2} \\ x_{2} + x_{3} \\ x_{3} + x_{4} \end{matrix}]$

Claim: $T$ is not one-one.

Consider $T ([\begin{matrix} 1 \\ 2 \\ 3 \\ 4 \end{matrix}]) = [\begin{matrix} 0 \\ 3 \\ 5 \\ 7 \end{matrix}]$ and $T ([\begin{matrix} 2 \\ 1 \\ 4 \\ 3 \end{matrix}]) = [\begin{matrix} 0 \\ 3 \\ 5 \\ 7 \end{matrix}]$

but $[\begin{matrix} 1 \\ 2 \\ 3 \\ 4 \end{matrix}] \neq [\begin{matrix} 2 \\ 1 \\ 4 \\ 3 \end{matrix}]$ , Hence $T$ is not one-one.

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