Consider a 2 x 2 matrix 1 U 01 (1) Compute U², U³, and give an explicit formula for Un Problem 0.2 (2) Let Lu : R² → R² be the linear transformation given by Lu() Ur. For a given unit square S = {(x, y): 0 < x, y ≤ 1}, sketch the image Lu (S) of S. (3) Is U diagonalizable?
Consider a 2 x 2 matrix 1 U 01 (1) Compute U², U³, and give an explicit formula for Un Problem 0.2 (2) Let Lu : R² → R² be the linear transformation given by Lu() Ur. For a given unit square S = {(x, y): 0 < x, y ≤ 1}, sketch the image Lu (S) of S. (3) Is U diagonalizable?
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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![**Problem 0.2**
Consider a \(2 \times 2\) matrix
\[
U = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}.
\]
1. Compute \(U^2\), \(U^3\), and give an explicit formula for \(U^n\).
2. Let \(L_U : \mathbb{R}^2 \rightarrow \mathbb{R}^2\) be the linear transformation given by \(L_U(\vec{x}) = U \vec{x}\). For a given unit square
\[
S = \{(x, y) : 0 \leq x, y \leq 1\},
\]
sketch the image \(L_U(S)\) of \(S\).
3. Is \(U\) diagonalizable?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3ef06bb3-2d9b-4f27-bb3b-835b443ab608%2Ff4d6318e-a55a-4ab3-b77b-61566be8380c%2F2o3cg5j_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 0.2**
Consider a \(2 \times 2\) matrix
\[
U = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}.
\]
1. Compute \(U^2\), \(U^3\), and give an explicit formula for \(U^n\).
2. Let \(L_U : \mathbb{R}^2 \rightarrow \mathbb{R}^2\) be the linear transformation given by \(L_U(\vec{x}) = U \vec{x}\). For a given unit square
\[
S = \{(x, y) : 0 \leq x, y \leq 1\},
\]
sketch the image \(L_U(S)\) of \(S\).
3. Is \(U\) diagonalizable?
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