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T F If A is m x n matrix, then ker(A) = ker(AT A).

T F If A is m x n matrix, then ker(A) = ker(AT A).

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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### True/False Question: (iii) **T** **F** \_\_\_. If \( A \) is an \( m \times n \) matrix, then \( \text{ker}(A) = \text{ker}(A^T A) \). **Explanation:** This question provides a statement about the relationship between a matrix \( A \) and its transpose multiplied by itself, specifically focusing on their kernels (null spaces). The question asks if the kernel (null space) of \( A \) is equal to the kernel of \( A^T A \), where \( A^T \) represents the transpose of the matrix \( A \). ### Understanding Kernels: - **Kernel of \( A \):** The set of all vectors \( \mathbf{x} \) such that \( A \mathbf{x} = \mathbf{0} \). - **Kernel of \( A^T A \):** The set of all vectors \( \mathbf{x} \) such that \( A^T A \mathbf{x} = \mathbf{0} \). The answer to the question depends on your understanding of linear algebra, specifically the properties of matrices and their kernels.
Transcribed Image Text:### True/False Question: (iii) **T** **F** \_\_\_. If \( A \) is an \( m \times n \) matrix, then \( \text{ker}(A) = \text{ker}(A^T A) \). **Explanation:** This question provides a statement about the relationship between a matrix \( A \) and its transpose multiplied by itself, specifically focusing on their kernels (null spaces). The question asks if the kernel (null space) of \( A \) is equal to the kernel of \( A^T A \), where \( A^T \) represents the transpose of the matrix \( A \). ### Understanding Kernels: - **Kernel of \( A \):** The set of all vectors \( \mathbf{x} \) such that \( A \mathbf{x} = \mathbf{0} \). - **Kernel of \( A^T A \):** The set of all vectors \( \mathbf{x} \) such that \( A^T A \mathbf{x} = \mathbf{0} \). The answer to the question depends on your understanding of linear algebra, specifically the properties of matrices and their kernels.
1. Determine whether each statement is true or false, and justify your answer.
Transcribed Image Text:1. Determine whether each statement is true or false, and justify your answer.
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