Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN: 9780134463216
Author: Robert F. Blitzer
Publisher: PEARSON
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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). --- **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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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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