A linear transformation T : R" → R" is invertibe if there exists a transformation S : R" → R" withS(T(7)) = T(S()) = 7 for all 7 e R". In this case S is called the inverse of T, and written as T1. In otherwords, T–1 will map the image of some vector a back to itself (note that this is just like regular functions).0 01 0-1 -3 2 11(a) If T: R" → R" is the matrix transformation T(7) = Ai where A =find the rule for the transformation T-1(b) Show that your transformation works by calculating T(B) and T-1(T(B)) if b =124

A linear transformation T:ℝn→ℝnis invertible if there exists a transformation S:ℝn→ℝn with…

Answered: A lincar transformation T : R" → R" is…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

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A linear transformation T : R" → R" is invertibe if there exists a transformation S : R" → R" with
S(T(7)) = T(S()) = 7 for all 7 e R". In this case S is called the inverse of T, and written as T1. In other
words, T–1 will map the image of some vector a back to itself (note that this is just like regular functions).
0 0
1 0
-1 -3 2 1
1
(a) If T: R" → R" is the matrix transformation T(7) = Ai where A =
find the rule for the transformation T-1
(b) Show that your transformation works by calculating T(B) and T-1(T(B)) if b =
124

Expert Solution

Step 1

A linear transformation $T : ℝ^{n} \to ℝ^{n}$ is invertible if there exists a transformation $S : ℝ^{n} \to ℝ^{n}$ with $S (T (Z)) = T (S (Z)) = Z \forall Z \in ℝ^{n}$ This S is known as inverse of linear transformation T.

(a)Given a linear transformation $T : ℝ^{4} \to ℝ^{4}$ such that $T (Z) = A Z$ where $A = [\begin{matrix} 1 & 0 & 0 & 0 \\ 2 & 1 & 0 & 0 \\ 4 & 2 & 1 & 0 \\ - 1 & - 3 & 2 & 1 \end{matrix}]$ .

Let $Z = [\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{matrix}]$ $Y = [\begin{matrix} y_{1} \\ y_{2} \\ y_{3} \\ y_{4} \end{matrix}]$ .

We need to find Y in terms of Z.

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