Use of computer is allowed Determine if the given linear transformation T:R → R° is one-to-one:|w = x, - 2.x, + 2.x, – x, +3x5w, = 2x, +x, + X3 + X4 – X5+x3 - X4— Зх, + 2х, -Зх,W; = X,W4||[Ws =2x, +4x,— 3х, + Xs

Answered: Determine if the given linear…

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

Use of computer is allowed

Determine if the given linear transformation T:R → R° is one-to-one:
|w = x, - 2.x, + 2.x, – x, +3x5
w, = 2x, +x, + X3 + X4 – X5
+x3 - X4
— Зх, + 2х, -Зх,
W; = X,
W4
||
[Ws =2x, +4x,
— 3х, + Xs

Expert Solution

Step 1

We can write the given system of equations in the matrix form as below:

$[\begin{matrix} w_{1} \\ w_{2} \\ w_{3} \\ w_{4} \\ w_{5} \end{matrix}] = [\begin{matrix} 1 & - 2 & 2 & - 1 & 3 \\ 2 & 1 & 1 & 1 & - 5 \\ 1 & 0 & 1 & - 1 & 0 \\ 0 & - 3 & 2 & 0 & - 3 \\ 2 & 4 & 0 & - 3 & 1 \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \\ x_{5} \end{matrix}]$

We have to determine whether the given transformation is linear or not.

Consider the matrix $A = [\begin{matrix} 1 & - 2 & 2 & - 1 & 3 \\ 2 & 1 & 1 & 1 & - 5 \\ 1 & 0 & 1 & - 1 & 0 \\ 0 & - 3 & 2 & 0 & - 3 \\ 2 & 4 & 0 & - 3 & 1 \end{matrix}]$ .