### 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

This question is about application of linear algebra

Answered: For the matrix A below, find a nonzero…

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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Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Question

### 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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