Let W be the subspace spanned by Find a basis for W and the dimension of W. -3 (Note that the reduced row echelon form of (a) Basis: (b) Dimension: 1 -3 -2 3 -3 -9 -6 1 -1 1 -1 -3 3 3 -3 1 0 4 -1 Is 3 3 0 10 0 30-10 0 1 -20 00 0 1 000 10
Let W be the subspace spanned by Find a basis for W and the dimension of W. -3 (Note that the reduced row echelon form of (a) Basis: (b) Dimension: 1 -3 -2 3 -3 -9 -6 1 -1 1 -1 -3 3 3 -3 1 0 4 -1 Is 3 3 0 10 0 30-10 0 1 -20 00 0 1 000 10
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter4: Vector Spaces
Section4.3: Subspaces Of Vector Spaces
Problem 56E: Give an example showing that the union of two subspaces of a vector space V is not necessarily a...
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![**Subspaces and Basis**
Let \( W \) be the subspace spanned by
\[ \left\{ \begin{bmatrix} 1 \\ -1 \\ 3 \\ -2 \end{bmatrix}, \begin{bmatrix} 3 \\ 3 \\ -9 \\ -6 \end{bmatrix}, \begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 3 \\ 3 \\ 0 \\ 4 \end{bmatrix}, \begin{bmatrix} 3 \\ 3 \\ 0 \\ -1 \end{bmatrix} \right\} \]
Find a basis for \( W \) and the dimension of \( W \).
(Note that the reduced row echelon form of
\[ \begin{bmatrix} 1 & 3 & 1 & 3 & 3 \\ -1 & 3 & 1 & 3 & 3 \\ 3 & -9 & 1 & 0 & 0 \\ -2 & 6 & -1 & 4 & -1 \end{bmatrix} \]
is
\[ \begin{bmatrix} 1 & 3 & 0 & -1 & 0 \\ 0 & 0 & 1 & 2 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix} \])
**(a) Basis:**
_Box for Basis_
**(b) Dimension:**
_Box for Dimension_
---
**Explanation of Row Echelon Form**
The given matrix and its reduced row echelon form (RREF) is key to solving this problem:
1. **Original Matrix**:
\[ \begin{bmatrix} 1 & 3 & 1 & 3 & 3 \\ -1 & 3 & 1 & 3 & 3 \\ 3 & -9 & 1 & 0 & 0 \\ -2 & 6 & -1 & 4 & -1 \end{bmatrix} \]
2. **Reduced Row Echelon Form (RREF)**:
\[ \begin{bmatrix} 1 & 3 & 0 & -1 & 0 \\ 0 & 0](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffb2133c9-e1e5-4d56-9c72-044227328930%2Ff5327d05-0073-4c8b-a5da-be820229d896%2Fharbp44_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Subspaces and Basis**
Let \( W \) be the subspace spanned by
\[ \left\{ \begin{bmatrix} 1 \\ -1 \\ 3 \\ -2 \end{bmatrix}, \begin{bmatrix} 3 \\ 3 \\ -9 \\ -6 \end{bmatrix}, \begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 3 \\ 3 \\ 0 \\ 4 \end{bmatrix}, \begin{bmatrix} 3 \\ 3 \\ 0 \\ -1 \end{bmatrix} \right\} \]
Find a basis for \( W \) and the dimension of \( W \).
(Note that the reduced row echelon form of
\[ \begin{bmatrix} 1 & 3 & 1 & 3 & 3 \\ -1 & 3 & 1 & 3 & 3 \\ 3 & -9 & 1 & 0 & 0 \\ -2 & 6 & -1 & 4 & -1 \end{bmatrix} \]
is
\[ \begin{bmatrix} 1 & 3 & 0 & -1 & 0 \\ 0 & 0 & 1 & 2 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix} \])
**(a) Basis:**
_Box for Basis_
**(b) Dimension:**
_Box for Dimension_
---
**Explanation of Row Echelon Form**
The given matrix and its reduced row echelon form (RREF) is key to solving this problem:
1. **Original Matrix**:
\[ \begin{bmatrix} 1 & 3 & 1 & 3 & 3 \\ -1 & 3 & 1 & 3 & 3 \\ 3 & -9 & 1 & 0 & 0 \\ -2 & 6 & -1 & 4 & -1 \end{bmatrix} \]
2. **Reduced Row Echelon Form (RREF)**:
\[ \begin{bmatrix} 1 & 3 & 0 & -1 & 0 \\ 0 & 0
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