Suppose a subspace is spanned by the set of vectors shown. Find a basis for the subspace, using the method of transforming a matrix to echelon form, where the rows of the matrix represent vectors spanning the subspace. (0-1²4]} (88) What is the dimension of the subspace?
Suppose a subspace is spanned by the set of vectors shown. Find a basis for the subspace, using the method of transforming a matrix to echelon form, where the rows of the matrix represent vectors spanning the subspace. (0-1²4]} (88) What is the dimension of the subspace?
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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![Suppose a subspace is spanned by the set of vectors shown. Find a basis for the subspace, using the method of transforming a matrix to echelon form, where the rows of the matrix represent vectors spanning the subspace.
\[
\left\{
\begin{bmatrix}
7 \\
9
\end{bmatrix},
\begin{bmatrix}
21 \\
-2
\end{bmatrix}
\right\}
\cdot
\begin{bmatrix}
[ \quad {} \quad ] \\
[ \quad {} \quad ]
\end{bmatrix}
\]
What is the dimension of the subspace?
**Explanation:**
- The set consists of two vectors: \(\begin{bmatrix} 7 \\ 9 \end{bmatrix}\) and \(\begin{bmatrix} 21 \\ -2 \end{bmatrix}\).
- To find a basis, form a matrix with these vectors as rows and transform it to echelon form.
- Determine the pivot positions in the matrix to identify the basis vectors.
- The dimension of the subspace is the number of pivot positions, representing the number of linearly independent vectors in the set.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc167378a-6772-4355-bdb0-a5fec1af8d0e%2F4452a354-c078-46a5-8f49-bbe6745cd6db%2Fxhlk42f_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Suppose a subspace is spanned by the set of vectors shown. Find a basis for the subspace, using the method of transforming a matrix to echelon form, where the rows of the matrix represent vectors spanning the subspace.
\[
\left\{
\begin{bmatrix}
7 \\
9
\end{bmatrix},
\begin{bmatrix}
21 \\
-2
\end{bmatrix}
\right\}
\cdot
\begin{bmatrix}
[ \quad {} \quad ] \\
[ \quad {} \quad ]
\end{bmatrix}
\]
What is the dimension of the subspace?
**Explanation:**
- The set consists of two vectors: \(\begin{bmatrix} 7 \\ 9 \end{bmatrix}\) and \(\begin{bmatrix} 21 \\ -2 \end{bmatrix}\).
- To find a basis, form a matrix with these vectors as rows and transform it to echelon form.
- Determine the pivot positions in the matrix to identify the basis vectors.
- The dimension of the subspace is the number of pivot positions, representing the number of linearly independent vectors in the set.
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