Let A = 10 0 1 1 -8 2 -1 1 -2 -21 2 -2 6 0 0 -2. and w - 13 2 0 2 [2] . Determine if w is in Col A. is w in Nul A?
Let A = 10 0 1 1 -8 2 -1 1 -2 -21 2 -2 6 0 0 -2. and w - 13 2 0 2 [2] . Determine if w is in Col A. is w in Nul A?
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter6: Linear Systems
Section6.3: Matrix Algebra
Problem 85E: Determine if the statement is true or false. If the statement is false, then correct it and make it...
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Determine if a
![### Linear Algebra Problem
#### Given Matrices:
Let \( A \) and \( w \) be matrices as defined below:
\[
A = \begin{bmatrix}
10 & -8 & -2 & -2 \\
0 & 2 & 2 & -2 \\
1 & -1 & 6 & 0 \\
1 & 1 & 0 & -2
\end{bmatrix}
\]
\[
w = \begin{bmatrix}
2 \\
2 \\
0 \\
2
\end{bmatrix}
\]
#### Problem Statement:
Determine if \( w \) is in \( \text{Col } A \). Is \( w \) in \( \text{Nul } A \)?
### Explanation
1. **Determine if \( w \) is in \( \text{Col } A \):**
To determine if the vector \( w \) is in the column space of \( A \), we need to check if there exists a vector \( x \) such that \( A x = w \). This involves solving the linear equation:
\[ A x = w \]
2. **Determine if \( w \) is in \( \text{Nul } A \):**
To determine if \( w \) is in the null space of \( A \), we need to check if \( A w = 0 \). This involves checking if multiplying the matrix \( A \) by the vector \( w \) results in the zero vector:
\[ A w = 0 \]
This matrix problem examines two fundamental concepts in linear algebra: the column space and the null space of a matrix. The column space (Col A) consists of all the linear combinations of the columns of \( A \), whereas the null space (Nul A) comprises all the solutions to the homogeneous equation \( A x = 0 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9baea4c0-9d9e-491a-8271-bc2a2bbd8b45%2Fec5d870b-1622-42d7-a7db-a41b9c39b1a5%2Fmt26nn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Linear Algebra Problem
#### Given Matrices:
Let \( A \) and \( w \) be matrices as defined below:
\[
A = \begin{bmatrix}
10 & -8 & -2 & -2 \\
0 & 2 & 2 & -2 \\
1 & -1 & 6 & 0 \\
1 & 1 & 0 & -2
\end{bmatrix}
\]
\[
w = \begin{bmatrix}
2 \\
2 \\
0 \\
2
\end{bmatrix}
\]
#### Problem Statement:
Determine if \( w \) is in \( \text{Col } A \). Is \( w \) in \( \text{Nul } A \)?
### Explanation
1. **Determine if \( w \) is in \( \text{Col } A \):**
To determine if the vector \( w \) is in the column space of \( A \), we need to check if there exists a vector \( x \) such that \( A x = w \). This involves solving the linear equation:
\[ A x = w \]
2. **Determine if \( w \) is in \( \text{Nul } A \):**
To determine if \( w \) is in the null space of \( A \), we need to check if \( A w = 0 \). This involves checking if multiplying the matrix \( A \) by the vector \( w \) results in the zero vector:
\[ A w = 0 \]
This matrix problem examines two fundamental concepts in linear algebra: the column space and the null space of a matrix. The column space (Col A) consists of all the linear combinations of the columns of \( A \), whereas the null space (Nul A) comprises all the solutions to the homogeneous equation \( A x = 0 \).
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