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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![### Linear Algebra: Checking if a Vector is in the Null Space
**Problem Statement:**
Given the matrix \( A \) and vector \( \mathbf{v} \), determine if \( \mathbf{v} \) is in the null space of \( A \).
Matrix \( A \) is defined as:
\[
A = \begin{bmatrix} 1 & 0 & -1 \\ 1 & 1 & 1 \end{bmatrix}
\]
Vector \( \mathbf{v} \) is:
\[
\mathbf{v} = \begin{bmatrix} -1 \\ 5 \\ -1 \end{bmatrix}
\]
**Question:**
Is \( \mathbf{v} \) in \( \text{null}(A) \)?
- ☐ \( \mathbf{v} \) is in \( \text{null}(A) \).
- ☐ \( \mathbf{v} \) is not in \( \text{null}(A) \).
**Explanation:**
To determine if \( \mathbf{v} \) is in \( \text{null}(A) \), compute \( A \mathbf{v} \). If the result is the zero vector, \( \mathbf{v} \) is in the null space of \( A \). Otherwise, it is not.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff6948d5d-782c-4f32-af22-abf47b4c3397%2Fff9c3977-4459-4335-ba29-d91a746b0e1a%2Fm3dex1b_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Linear Algebra: Checking if a Vector is in the Null Space
**Problem Statement:**
Given the matrix \( A \) and vector \( \mathbf{v} \), determine if \( \mathbf{v} \) is in the null space of \( A \).
Matrix \( A \) is defined as:
\[
A = \begin{bmatrix} 1 & 0 & -1 \\ 1 & 1 & 1 \end{bmatrix}
\]
Vector \( \mathbf{v} \) is:
\[
\mathbf{v} = \begin{bmatrix} -1 \\ 5 \\ -1 \end{bmatrix}
\]
**Question:**
Is \( \mathbf{v} \) in \( \text{null}(A) \)?
- ☐ \( \mathbf{v} \) is in \( \text{null}(A) \).
- ☐ \( \mathbf{v} \) is not in \( \text{null}(A) \).
**Explanation:**
To determine if \( \mathbf{v} \) is in \( \text{null}(A) \), compute \( A \mathbf{v} \). If the result is the zero vector, \( \mathbf{v} \) is in the null space of \( A \). Otherwise, it is not.
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