nd a basis for the subspace of R³ spanned by S. S = {(5, 5, 9), (1, 1, 3), (1, 1, 1)} TEP 1: Find the reduced row-echelon form of the matrix whose rows are the vectors in S. 000 TEP 2: Determine a basis that spans S. 183
nd a basis for the subspace of R³ spanned by S. S = {(5, 5, 9), (1, 1, 3), (1, 1, 1)} TEP 1: Find the reduced row-echelon form of the matrix whose rows are the vectors in S. 000 TEP 2: Determine a basis that spans S. 183
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![**Finding a Basis for a Subspace of \( \mathbb{R}^3 \) Spanned by a Set**
*Objective:* Find a basis for the subspace of \( \mathbb{R}^3 \) spanned by the set \( S = \{(5, 5, 9), (1, 1, 3), (1, 1, 1)\} \).
**STEP 1:** Find the reduced row-echelon form of the matrix whose rows are the vectors in \( S \).
Matrix Representation:
\[
\begin{bmatrix}
5 & 5 & 9 \\
1 & 1 & 3 \\
1 & 1 & 1 \\
\end{bmatrix}
\]
- Perform row operations to transform this matrix into its reduced row-echelon form.
**STEP 2:** Determine a basis that spans \( S \).
- Extract the non-zero rows from the reduced row-echelon form to form a basis for the subspace.
**Explanation of the Process:**
1. **Matrix Formation**: Arrange the vectors in \( S \) as rows of a matrix.
2. **Row Reduction**: Use elementary row operations (swap, scale, replace) to convert the matrix into its reduced row-echelon form.
3. **Basis Determination**: The non-zero rows in the reduced form will provide a basis for the subspace spanned by \( S \).
Note: The arrows indicate the process flow from forming the matrix to obtaining a basis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9348d578-66ea-4064-b97e-301aeae52239%2F10bf39a2-3ef3-41e2-93fc-4a816490393c%2F0hren7oi_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Finding a Basis for a Subspace of \( \mathbb{R}^3 \) Spanned by a Set**
*Objective:* Find a basis for the subspace of \( \mathbb{R}^3 \) spanned by the set \( S = \{(5, 5, 9), (1, 1, 3), (1, 1, 1)\} \).
**STEP 1:** Find the reduced row-echelon form of the matrix whose rows are the vectors in \( S \).
Matrix Representation:
\[
\begin{bmatrix}
5 & 5 & 9 \\
1 & 1 & 3 \\
1 & 1 & 1 \\
\end{bmatrix}
\]
- Perform row operations to transform this matrix into its reduced row-echelon form.
**STEP 2:** Determine a basis that spans \( S \).
- Extract the non-zero rows from the reduced row-echelon form to form a basis for the subspace.
**Explanation of the Process:**
1. **Matrix Formation**: Arrange the vectors in \( S \) as rows of a matrix.
2. **Row Reduction**: Use elementary row operations (swap, scale, replace) to convert the matrix into its reduced row-echelon form.
3. **Basis Determination**: The non-zero rows in the reduced form will provide a basis for the subspace spanned by \( S \).
Note: The arrows indicate the process flow from forming the matrix to obtaining a basis.
Expert Solution
Step 1
Given:
Set of vectors
To find:
Determine the basis for the subspace of spanned by .
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