For the matrix A below, find a nonzero vector in Nul A and a nonzero vector in Col A. 2 3 8 - 11 A= -8 - 9 - 20 29 8 6 8 - 14 Find a nonzero vector in Nul A.
For the matrix A below, find a nonzero vector in Nul A and a nonzero vector in Col A. 2 3 8 - 11 A= -8 - 9 - 20 29 8 6 8 - 14 Find a nonzero vector in Nul A.
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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![### Finding Nonzero Vectors in Null Space and Column Space of a Matrix
#### Problem Statement:
For the matrix \( A \) below, find a nonzero vector in \( \text{Nul}\,A \) (Null Space of A) and a nonzero vector in \( \text{Col}\,A \) (Column Space of A).
Given matrix:
\[
A = \begin{bmatrix}
2 & 3 & 8 & -11 \\
-8 & -9 & -20 & 29 \\
8 & 6 & 8 & -14 \\
\end{bmatrix}
\]
#### Tasks:
- Find a nonzero vector in \( \text{Nul}\,A \).
- Find a nonzero vector in \( \text{Col}\,A \).
### Solution Steps:
**1. Find a Nonzero Vector in \( \text{Nul}\,A \):**
To find a vector in \( \text{Nul}\,A \), we need to solve the equation \( A \mathbf{x} = \mathbf{0} \), where \(\mathbf{x}\) is a vector and \(\mathbf{0}\) is the zero vector.
Let's denote \(\mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix}\).
So, we have:
\[
A \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix} = \begin{bmatrix} 2x_1 + 3x_2 + 8x_3 - 11x_4 \\ -8x_1 - 9x_2 - 20x_3 + 29x_4 \\ 8x_1 + 6x_2 + 8x_3 - 14x_4 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}
\]
By solving this system of linear equations, we can determine the necessary values of \(x_1, x_2, x_3,\) and \(x_4\) that satisfy the equation.
**2. Find a Nonzero Vector in \( \text{Col}\,A \):**
The column space of \(A](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F965f2633-6838-4c2d-b94e-32b85de15479%2Faf22f1ca-6395-4e7c-aaf6-8bae85d44ddc%2Fbu50nmd_processed.png&w=3840&q=75)
Transcribed Image Text:### Finding Nonzero Vectors in Null Space and Column Space of a Matrix
#### Problem Statement:
For the matrix \( A \) below, find a nonzero vector in \( \text{Nul}\,A \) (Null Space of A) and a nonzero vector in \( \text{Col}\,A \) (Column Space of A).
Given matrix:
\[
A = \begin{bmatrix}
2 & 3 & 8 & -11 \\
-8 & -9 & -20 & 29 \\
8 & 6 & 8 & -14 \\
\end{bmatrix}
\]
#### Tasks:
- Find a nonzero vector in \( \text{Nul}\,A \).
- Find a nonzero vector in \( \text{Col}\,A \).
### Solution Steps:
**1. Find a Nonzero Vector in \( \text{Nul}\,A \):**
To find a vector in \( \text{Nul}\,A \), we need to solve the equation \( A \mathbf{x} = \mathbf{0} \), where \(\mathbf{x}\) is a vector and \(\mathbf{0}\) is the zero vector.
Let's denote \(\mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix}\).
So, we have:
\[
A \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix} = \begin{bmatrix} 2x_1 + 3x_2 + 8x_3 - 11x_4 \\ -8x_1 - 9x_2 - 20x_3 + 29x_4 \\ 8x_1 + 6x_2 + 8x_3 - 14x_4 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}
\]
By solving this system of linear equations, we can determine the necessary values of \(x_1, x_2, x_3,\) and \(x_4\) that satisfy the equation.
**2. Find a Nonzero Vector in \( \text{Col}\,A \):**
The column space of \(A
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