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,...
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Question
![**Title:** Finding Vectors that Span the Kernel of a Matrix
**Problem Statement:**
Determine vectors that span the kernel of matrix \( A \).
**Matrix \( A \):**
\[
A = \begin{bmatrix}
1 & 1 & 1 \\
1 & 2 & 5 \\
1 & 3 & 9
\end{bmatrix}
\]
**Explanation:**
To find the vectors that span the kernel (null space) of matrix \( A \), we need to solve the equation:
\[ A \mathbf{x} = \mathbf{0} \]
where \( \mathbf{x} \) is a vector and \( \mathbf{0} \) is the zero vector. The solution will provide a set of vectors that, when multiplied by \( A \), result in the zero vector, representing the subspace that forms the kernel.
**Steps:**
1. **Row Reduction:** Perform row operations to reduce matrix \( A \) to its row echelon form or reduced row echelon form.
2. **Solving the System:** Look for the solution set of the system of linear equations in \( A \mathbf{x} = \mathbf{0} \).
3. **Span Description:** Express the solution in terms of free variables, which give the spanning vectors for the kernel.
Understanding and finding the vectors that span the kernel helps in various applications such as solving linear equations, simplifying systems, and understanding the structure of a matrix in linear algebra.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8cf9958e-dd38-41eb-b691-55a1e2252bf8%2F5779f197-ef13-4ece-a6bc-ea0761e39f58%2Firfm24i_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title:** Finding Vectors that Span the Kernel of a Matrix
**Problem Statement:**
Determine vectors that span the kernel of matrix \( A \).
**Matrix \( A \):**
\[
A = \begin{bmatrix}
1 & 1 & 1 \\
1 & 2 & 5 \\
1 & 3 & 9
\end{bmatrix}
\]
**Explanation:**
To find the vectors that span the kernel (null space) of matrix \( A \), we need to solve the equation:
\[ A \mathbf{x} = \mathbf{0} \]
where \( \mathbf{x} \) is a vector and \( \mathbf{0} \) is the zero vector. The solution will provide a set of vectors that, when multiplied by \( A \), result in the zero vector, representing the subspace that forms the kernel.
**Steps:**
1. **Row Reduction:** Perform row operations to reduce matrix \( A \) to its row echelon form or reduced row echelon form.
2. **Solving the System:** Look for the solution set of the system of linear equations in \( A \mathbf{x} = \mathbf{0} \).
3. **Span Description:** Express the solution in terms of free variables, which give the spanning vectors for the kernel.
Understanding and finding the vectors that span the kernel helps in various applications such as solving linear equations, simplifying systems, and understanding the structure of a matrix in linear algebra.
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convert the given matrix into RREF
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