(2) Suppose that V is a finite dimensional vector space over R. Show that if dim(V) is odd, then every T E L(V) has an eigenvalue. (Hint: use induction).
(2) Suppose that V is a finite dimensional vector space over R. Show that if dim(V) is odd, then every T E L(V) has an eigenvalue. (Hint: use induction).
Algebra and Trigonometry (6th Edition)
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ChapterP: Prerequisites: Fundamental Concepts Of Algebra
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Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![**Problem 2:** Suppose that \( V \) is a finite dimensional vector space over \( \mathbb{R} \). Show that if \( \text{dim}(V) \) is odd, then every \( T \in \mathcal{L}(V) \) has an eigenvalue. (Hint: use induction).
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**Explanation for Educational Context:**
This problem is focused on the properties of linear operators on vector spaces. The goal is to demonstrate that if a vector space \( V \) over the real numbers \( \mathbb{R} \) has an odd dimension, then any linear transformation \( T \) from \( V \) to itself (denoted by \( T \in \mathcal{L}(V) \)) possesses at least one eigenvalue. The hint suggests employing mathematical induction as a method to prove this statement.
**Key Concepts:**
- **Vector Space \( V \):** A collection of vectors where vector addition and scalar multiplication are defined.
- **Dimension \( \text{dim}(V) \):** The number of vectors in a basis of the vector space, indicating its size or complexity.
- **Linear Transformation \( T \):** A function from a vector space to itself that preserves vector addition and scalar multiplication.
- **Eigenvalue:** A scalar \( \lambda \) such that there exists a nonzero vector \( v \) in \( V \) where \( T(v) = \lambda v \).
- **Induction:** A mathematical technique often used to prove statements about all natural numbers, particularly useful for proving properties of dimensions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F36d93af9-1c60-4c54-a265-7600bbfc6b68%2F4fa63e59-1757-4255-aefe-7d2dcf3dde7c%2F0xuip8d_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 2:** Suppose that \( V \) is a finite dimensional vector space over \( \mathbb{R} \). Show that if \( \text{dim}(V) \) is odd, then every \( T \in \mathcal{L}(V) \) has an eigenvalue. (Hint: use induction).
---
**Explanation for Educational Context:**
This problem is focused on the properties of linear operators on vector spaces. The goal is to demonstrate that if a vector space \( V \) over the real numbers \( \mathbb{R} \) has an odd dimension, then any linear transformation \( T \) from \( V \) to itself (denoted by \( T \in \mathcal{L}(V) \)) possesses at least one eigenvalue. The hint suggests employing mathematical induction as a method to prove this statement.
**Key Concepts:**
- **Vector Space \( V \):** A collection of vectors where vector addition and scalar multiplication are defined.
- **Dimension \( \text{dim}(V) \):** The number of vectors in a basis of the vector space, indicating its size or complexity.
- **Linear Transformation \( T \):** A function from a vector space to itself that preserves vector addition and scalar multiplication.
- **Eigenvalue:** A scalar \( \lambda \) such that there exists a nonzero vector \( v \) in \( V \) where \( T(v) = \lambda v \).
- **Induction:** A mathematical technique often used to prove statements about all natural numbers, particularly useful for proving properties of dimensions.
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