Homework Help is Here – Start Your Trial Now!
Math
Algebra
(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)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
See similar textbooks
icon
Related questions
Question

q2

**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.
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.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 2 steps with 1 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
Algebra
ISBN:
9780134463216
Author:
Robert F. Blitzer
Publisher:
PEARSON
Contemporary Abstract Algebra
Contemporary Abstract Algebra
Algebra
ISBN:
9781305657960
Author:
Joseph Gallian
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra And Trigonometry (11th Edition)
Algebra And Trigonometry (11th Edition)
Algebra
ISBN:
9780135163078
Author:
Michael Sullivan
Publisher:
PEARSON
Introduction to Linear Algebra, Fifth Edition
Introduction to Linear Algebra, Fifth Edition
Algebra
ISBN:
9780980232776
Author:
Gilbert Strang
Publisher:
Wellesley-Cambridge Press
College Algebra (Collegiate Math)
College Algebra (Collegiate Math)
Algebra
ISBN:
9780077836344
Author:
Julie Miller, Donna Gerken
Publisher:
McGraw-Hill Education
SEE MORE TEXTBOOKS