If the set W is a vector space, find a set S of vectors that spans it. Otherwise, state that W is not a vector space. W is the set of all vectors of the form O O O 155 6010 Not a vector space 69 05 6500 a +6b 5b 5a-b -a where a and b are arbitrary real numbers.
If the set W is a vector space, find a set S of vectors that spans it. Otherwise, state that W is not a vector space. W is the set of all vectors of the form O O O 155 6010 Not a vector space 69 05 6500 a +6b 5b 5a-b -a where a and b are arbitrary real numbers.
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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![**Vector Spaces**
**Problem Statement:**
If the set \( W \) is a vector space, find a set \( S \) of vectors that spans it. Otherwise, state that \( W \) is not a vector space.
\( W \) is the set of all vectors of the form:
\[ \begin{bmatrix} a + 6b \\ 5b \\ 5a - b \\ -a \end{bmatrix} \]
where \( a \) and \( b \) are arbitrary real numbers.
**Options:**
1.
\[ \left\{
\begin{bmatrix}
1 \\
0 \\
5 \\
0
\end{bmatrix},
\begin{bmatrix}
6 \\
5 \\
-1 \\
-1
\end{bmatrix}
\right\} \]
2.
\[ \left\{
\begin{bmatrix}
1 \\
0 \\
5 \\
-1
\end{bmatrix},
\begin{bmatrix}
6 \\
5 \\
-1 \\
0
\end{bmatrix}
\right\} \]
3.
\[ \left\{
\begin{bmatrix}
6 \\
5 \\
-1 \\
0
\end{bmatrix}
\right\} \]
4.
Not a vector space
Please choose the correct option from the above choices regarding the set \( W \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb47bac88-cba5-40e5-9f3f-db7316eb95a3%2F8502ea66-deec-4001-befc-f622b98116fd%2Fdxyej63_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Vector Spaces**
**Problem Statement:**
If the set \( W \) is a vector space, find a set \( S \) of vectors that spans it. Otherwise, state that \( W \) is not a vector space.
\( W \) is the set of all vectors of the form:
\[ \begin{bmatrix} a + 6b \\ 5b \\ 5a - b \\ -a \end{bmatrix} \]
where \( a \) and \( b \) are arbitrary real numbers.
**Options:**
1.
\[ \left\{
\begin{bmatrix}
1 \\
0 \\
5 \\
0
\end{bmatrix},
\begin{bmatrix}
6 \\
5 \\
-1 \\
-1
\end{bmatrix}
\right\} \]
2.
\[ \left\{
\begin{bmatrix}
1 \\
0 \\
5 \\
-1
\end{bmatrix},
\begin{bmatrix}
6 \\
5 \\
-1 \\
0
\end{bmatrix}
\right\} \]
3.
\[ \left\{
\begin{bmatrix}
6 \\
5 \\
-1 \\
0
\end{bmatrix}
\right\} \]
4.
Not a vector space
Please choose the correct option from the above choices regarding the set \( W \).
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