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Problem A. Consider the linear transformation T: R₁ [x] → R₁ [x] given by T(a + bx) = (a + b) + ax (A1) Is T cyclic? (A2) Is T irreducible? (A3) Is T indecomposable? I

Problem A. Consider the linear transformation T: R₁ [x] → R₁ [x] given by T(a + bx) = (a + b) + ax (A1) Is T cyclic? (A2) Is T irreducible? (A3) Is T indecomposable? I

Advanced Engineering Mathematics
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 A. Consider the linear transformation** \[ T: \mathbb{R}_1[x] \rightarrow \mathbb{R}_1[x] \] given by \[ T(a + bx) = (a + b) + ax \] (A1) Is \( T \) cyclic? (A2) Is \( T \) irreducible? (A3) Is \( T \) indecomposable? --- **Explanation:** - **Linear Transformation**: The function \( T \) is a linear transformation mapping polynomials of degree 1 to polynomials of degree 1. - **Definition**: \( T \) is defined by the rule \( T(a + bx) = (a + b) + ax \), where \( a \) and \( b \) are constants, and \( x \) is the variable. **Questions:** 1. **(A1) Cyclic**: - A transformation is cyclic if there exists a vector such that its cyclic subspace (generated by that vector) equals the entire space. 2. **(A2) Irreducible**: - A linear transformation is irreducible if there is no non-trivial invariant subspace that it leaves unchanged. 3. **(A3) Indecomposable**: - A transformation is indecomposable if it cannot be written as a direct sum of two or more invariant subspaces. --- There are no graphs or diagrams provided in the image. The text focuses solely on mathematical definitions and posing specific questions about the properties of the given linear transformation.
Transcribed Image Text:**Problem A. Consider the linear transformation** \[ T: \mathbb{R}_1[x] \rightarrow \mathbb{R}_1[x] \] given by \[ T(a + bx) = (a + b) + ax \] (A1) Is \( T \) cyclic? (A2) Is \( T \) irreducible? (A3) Is \( T \) indecomposable? --- **Explanation:** - **Linear Transformation**: The function \( T \) is a linear transformation mapping polynomials of degree 1 to polynomials of degree 1. - **Definition**: \( T \) is defined by the rule \( T(a + bx) = (a + b) + ax \), where \( a \) and \( b \) are constants, and \( x \) is the variable. **Questions:** 1. **(A1) Cyclic**: - A transformation is cyclic if there exists a vector such that its cyclic subspace (generated by that vector) equals the entire space. 2. **(A2) Irreducible**: - A linear transformation is irreducible if there is no non-trivial invariant subspace that it leaves unchanged. 3. **(A3) Indecomposable**: - A transformation is indecomposable if it cannot be written as a direct sum of two or more invariant subspaces. --- There are no graphs or diagrams provided in the image. The text focuses solely on mathematical definitions and posing specific questions about the properties of the given linear transformation.
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