Advanced Engineering Mathematics
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ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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orthogonal vectors
![### Orthogonality and Linear Algebra: Problem 5
#### Problem Statement
Let \( A \) be an \( m \times n \) matrix and \( X \) an \( n \times 1 \) column vector. Consider the system:
\[ AX = 0 \]
1. **Explain why \( X \) is orthogonal to \( \text{Row}(A) \)**:
That is, it is orthogonal to every vector in \( \text{Row}(A) \).
*(Hint: It is enough if it is orthogonal to all vectors in a basis of \( \text{Row}(A) \).)*
2. **Consider the following matrix, which is already in reduced row echelon form (RREF):**
\[
A = \begin{bmatrix}
1 & 0 & -2 & 2 \\
0 & 1 & 1 & -3 \\
0 & 0 & 1 & -4
\end{bmatrix}
\]
Find one vector that is orthogonal to \( \text{Row}(A) \).
*(Just one vector is enough.)*
#### Solution
**Solution to Problem 5:**
---
**Additional Explanation:**
1. For the matrix \( A \) given:
\[ A = \begin{bmatrix}
1 & 0 & -2 & 2 \\
0 & 1 & 1 & -3 \\
0 & 0 & 1 & -4
\end{bmatrix} \]
The problem asks us to find a vector \( X \) such that:
\[ AX = 0 \]
This will show that \( X \) is orthogonal to every row of \( A \), hence to \( \text{Row}(A) \), because any such \( X \) will be orthogonal to the span of the rows of \( A \).
---](https://content.bartleby.com/qna-images/question/be52e616-7bd7-4d85-a476-7e5fcec53ee5/17e386e5-f7bc-4146-83b2-576c09c65c6e/0b37tlo_processed.jpeg)
Transcribed Image Text:### Orthogonality and Linear Algebra: Problem 5
#### Problem Statement
Let \( A \) be an \( m \times n \) matrix and \( X \) an \( n \times 1 \) column vector. Consider the system:
\[ AX = 0 \]
1. **Explain why \( X \) is orthogonal to \( \text{Row}(A) \)**:
That is, it is orthogonal to every vector in \( \text{Row}(A) \).
*(Hint: It is enough if it is orthogonal to all vectors in a basis of \( \text{Row}(A) \).)*
2. **Consider the following matrix, which is already in reduced row echelon form (RREF):**
\[
A = \begin{bmatrix}
1 & 0 & -2 & 2 \\
0 & 1 & 1 & -3 \\
0 & 0 & 1 & -4
\end{bmatrix}
\]
Find one vector that is orthogonal to \( \text{Row}(A) \).
*(Just one vector is enough.)*
#### Solution
**Solution to Problem 5:**
---
**Additional Explanation:**
1. For the matrix \( A \) given:
\[ A = \begin{bmatrix}
1 & 0 & -2 & 2 \\
0 & 1 & 1 & -3 \\
0 & 0 & 1 & -4
\end{bmatrix} \]
The problem asks us to find a vector \( X \) such that:
\[ AX = 0 \]
This will show that \( X \) is orthogonal to every row of \( A \), hence to \( \text{Row}(A) \), because any such \( X \) will be orthogonal to the span of the rows of \( A \).
---
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