2. Let T : R³ → R³ be the operator given by -1 1 0 0 −1 1 0 0 – 1 - T(v) = = v. Determine whether T is decomposable or indecomposable.
2. Let T : R³ → R³ be the operator given by -1 1 0 0 −1 1 0 0 – 1 - T(v) = = v. Determine whether T is decomposable or indecomposable.
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 2: Linear Algebra - Decomposability of a Linear Operator
---
**Problem Statement:**
2. Let \( T : \mathbb{R}^3 \rightarrow \mathbb{R}^3 \) be the operator given by
\[ T(v) = \begin{pmatrix}
-1 & 1 & 0 \\
0 & -1 & 1 \\
0 & 0 & -1
\end{pmatrix} v. \]
Determine whether \( T \) is decomposable or indecomposable.
---
**Explanation:**
In this problem, we are given a linear operator \( T \) which maps vectors from the 3-dimensional real vector space \( \mathbb{R}^3 \) back to \( \mathbb{R}^3 \). The operator \( T \) is represented by a \( 3 \times 3 \) matrix:
\[ T = \begin{pmatrix}
-1 & 1 & 0 \\
0 & -1 & 1 \\
0 & 0 & -1
\end{pmatrix}. \]
The task is to determine whether the operator \( T \) is decomposable or indecomposable. A linear operator is said to be decomposable if the vector space \( \mathbb{R}^3 \) can be written as a direct sum of two nontrivial \( T \)-invariant subspaces. Otherwise, \( T \) is indecomposable.
---
For a deeper understanding, you can:
1. **Examine the Matrix:** Look at its eigenvalues and eigenvectors to determine if there are any invariant subspaces.
2. **Check for Diagonalizability:** If the matrix can be diagonalized, it might be decomposable.
3. **Geometric Multiplicity:** Inspect the geometric multiplicity of each eigenvalue. If it is less than the algebraic multiplicity, the matrix might be indecomposable.
To solve this specific problem, one could start by finding the eigenvalues and eigenvectors of the matrix representing \( T \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fae19e8bd-5915-473c-8a03-9ac053dffb7e%2Fbeed83fc-2d27-40ce-a605-26cb295dc15b%2Fmgisgdg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem 2: Linear Algebra - Decomposability of a Linear Operator
---
**Problem Statement:**
2. Let \( T : \mathbb{R}^3 \rightarrow \mathbb{R}^3 \) be the operator given by
\[ T(v) = \begin{pmatrix}
-1 & 1 & 0 \\
0 & -1 & 1 \\
0 & 0 & -1
\end{pmatrix} v. \]
Determine whether \( T \) is decomposable or indecomposable.
---
**Explanation:**
In this problem, we are given a linear operator \( T \) which maps vectors from the 3-dimensional real vector space \( \mathbb{R}^3 \) back to \( \mathbb{R}^3 \). The operator \( T \) is represented by a \( 3 \times 3 \) matrix:
\[ T = \begin{pmatrix}
-1 & 1 & 0 \\
0 & -1 & 1 \\
0 & 0 & -1
\end{pmatrix}. \]
The task is to determine whether the operator \( T \) is decomposable or indecomposable. A linear operator is said to be decomposable if the vector space \( \mathbb{R}^3 \) can be written as a direct sum of two nontrivial \( T \)-invariant subspaces. Otherwise, \( T \) is indecomposable.
---
For a deeper understanding, you can:
1. **Examine the Matrix:** Look at its eigenvalues and eigenvectors to determine if there are any invariant subspaces.
2. **Check for Diagonalizability:** If the matrix can be diagonalized, it might be decomposable.
3. **Geometric Multiplicity:** Inspect the geometric multiplicity of each eigenvalue. If it is less than the algebraic multiplicity, the matrix might be indecomposable.
To solve this specific problem, one could start by finding the eigenvalues and eigenvectors of the matrix representing \( T \).
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