STEP 1: Find the reduced row-echelon form of the matrix whose rows are the vectors in S. 00
STEP 1: Find the reduced row-echelon form of the matrix whose rows are the vectors in S. 00
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Finding a Basis for the Subspace of \( \mathbb{R}^3 \) Spanned by \( S \)
Given:
\[ S = \{ (5, 5, 9), (1, 1, 2), (1, 1, 1) \} \]
#### **Step 1: Find the Reduced Row-Echelon Form (RREF)**
To find the basis for the subspace of \( \mathbb{R}^3 \) spanned by \( S \), we need to find the reduced row-echelon form of the matrix formed by the vectors in \( S \).
The matrix whose rows are the vectors in \( S \) is:
\[
\begin{pmatrix}
5 & 5 & 9 \\
1 & 1 & 2 \\
1 & 1 & 1
\end{pmatrix}
\]
##### Explanation of Graphical Elements:
- **Matrix Setup:** The image contains a visual representation of a matrix setup. Each cell represents an element of the matrix, and it appears that there are arrows and outlines indicating manipulation of rows during the process of finding the RREF.
- **Empty Cells:** These boxes are placeholders for the elements of the matrix.
- **Arrows:**
- The green arrows pointing to the right and downward indicate the direction of some row operations to transform the matrix.
- There is a blue dashed outline around one cell, signifying the conversion or row operation centered around that pivot element.
In summary, proceed to use standard row operations (swap rows, multiply rows by a scalar, and add/subtract rows) to transform this matrix into its reduced row-echelon form.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc9c4e6d3-6045-4734-bab1-32fff54ce7c7%2Fd6256570-216b-46fb-9437-8a17ab1ac485%2Fb1ytceq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Finding a Basis for the Subspace of \( \mathbb{R}^3 \) Spanned by \( S \)
Given:
\[ S = \{ (5, 5, 9), (1, 1, 2), (1, 1, 1) \} \]
#### **Step 1: Find the Reduced Row-Echelon Form (RREF)**
To find the basis for the subspace of \( \mathbb{R}^3 \) spanned by \( S \), we need to find the reduced row-echelon form of the matrix formed by the vectors in \( S \).
The matrix whose rows are the vectors in \( S \) is:
\[
\begin{pmatrix}
5 & 5 & 9 \\
1 & 1 & 2 \\
1 & 1 & 1
\end{pmatrix}
\]
##### Explanation of Graphical Elements:
- **Matrix Setup:** The image contains a visual representation of a matrix setup. Each cell represents an element of the matrix, and it appears that there are arrows and outlines indicating manipulation of rows during the process of finding the RREF.
- **Empty Cells:** These boxes are placeholders for the elements of the matrix.
- **Arrows:**
- The green arrows pointing to the right and downward indicate the direction of some row operations to transform the matrix.
- There is a blue dashed outline around one cell, signifying the conversion or row operation centered around that pivot element.
In summary, proceed to use standard row operations (swap rows, multiply rows by a scalar, and add/subtract rows) to transform this matrix into its reduced row-echelon form.
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