h Let u = and v= Show that is in Span {u,v) for all h and k. How can it be shown that a vector b is in Span {u,v}? A. Determine if the system containing u, v, and b is consistent. If the system is consistent, b might be in Span (u,v). B. Determine if the system containing u, v, and b is consistent. If the system is consistent, then b is not in Span {u,v). C. Determine if the system containing u, v, and b is consistent. If the system is consistent, then b is in Span {u,v). O D. Determine if the system containing u, v, and b is consistent. If the system is inconsistent, then b is in Span (u,v). h Find the augmented matrix [u v b]. k Let b =
h Let u = and v= Show that is in Span {u,v) for all h and k. How can it be shown that a vector b is in Span {u,v}? A. Determine if the system containing u, v, and b is consistent. If the system is consistent, b might be in Span (u,v). B. Determine if the system containing u, v, and b is consistent. If the system is consistent, then b is not in Span {u,v). C. Determine if the system containing u, v, and b is consistent. If the system is consistent, then b is in Span {u,v). O D. Determine if the system containing u, v, and b is consistent. If the system is inconsistent, then b is in Span (u,v). h Find the augmented matrix [u v b]. k Let b =
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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![### Row Reduction and Consistency of Systems of Linear Equations
**Objective:**
Learn to row reduce the augmented matrix to its reduced echelon form and determine the consistency of the system for given variables \( h \) and \( k \).
**Given Augmented Matrix:**
\[
\left[\begin{array}{ccc|c}
11 & 11 & 1 & h \\
-1 & 1 & 1 & k \\
\end{array}\right]
\]
To proceed with row reduction, simplify this matrix into its reduced echelon form.
*After Row Reduction:*
(An example of a possible reduced echelon form result would be illustrated here, given proper steps.)
**Consistency Check:**
- **Dropdown Selection:** The system is [Select: consistent / inconsistent] for all values of \( h \) and \( k \).
- **Conclusion:** So it can be stated that
\[
\left[\begin{array}{c}
h \\
k \\
\end{array}\right]
\]
is in Span \(\{u,v\}\) for all values of \( h \) and \( k \).
**Concept Explanation:**
- **Row reduction** refers to the process of using elementary row operations to transform a given matrix into its row echelon form or reduced row echelon form.
- **Consistency** of a system is determined by whether the system of equations represented by the matrix has at least one solution (consistent) or no solutions (inconsistent).
- **Span** is the set of all possible linear combinations of a given set of vectors.
*Interactive Element:*
You can interact with the dropdown to select whether the system is “consistent” or “inconsistent” based on the values of \( h \) and \( k \).
By understanding the properties of row reduction and system consistency, you can solve and analyze various linear equations effectively.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc0867650-b445-489d-98f3-e27e64467d14%2Fd250f33a-b08a-4655-9489-c70032fde7c1%2Fq8fx5oc_processed.png&w=3840&q=75)
Transcribed Image Text:### Row Reduction and Consistency of Systems of Linear Equations
**Objective:**
Learn to row reduce the augmented matrix to its reduced echelon form and determine the consistency of the system for given variables \( h \) and \( k \).
**Given Augmented Matrix:**
\[
\left[\begin{array}{ccc|c}
11 & 11 & 1 & h \\
-1 & 1 & 1 & k \\
\end{array}\right]
\]
To proceed with row reduction, simplify this matrix into its reduced echelon form.
*After Row Reduction:*
(An example of a possible reduced echelon form result would be illustrated here, given proper steps.)
**Consistency Check:**
- **Dropdown Selection:** The system is [Select: consistent / inconsistent] for all values of \( h \) and \( k \).
- **Conclusion:** So it can be stated that
\[
\left[\begin{array}{c}
h \\
k \\
\end{array}\right]
\]
is in Span \(\{u,v\}\) for all values of \( h \) and \( k \).
**Concept Explanation:**
- **Row reduction** refers to the process of using elementary row operations to transform a given matrix into its row echelon form or reduced row echelon form.
- **Consistency** of a system is determined by whether the system of equations represented by the matrix has at least one solution (consistent) or no solutions (inconsistent).
- **Span** is the set of all possible linear combinations of a given set of vectors.
*Interactive Element:*
You can interact with the dropdown to select whether the system is “consistent” or “inconsistent” based on the values of \( h \) and \( k \).
By understanding the properties of row reduction and system consistency, you can solve and analyze various linear equations effectively.
![**Linear Algebra: Span and Consistency**
Let \( u = \begin{bmatrix} 11 \\ -1 \end{bmatrix} \) and \( v = \begin{bmatrix} 11 \\ 1 \end{bmatrix} \). Show that \( \begin{bmatrix} h \\ k \end{bmatrix} \) is in Span \(\{u,v\}\) for all \( h \) and \( k \).
**How can it be shown that a vector \( b \) is in Span \(\{u, v\}\)?**
- \( \quad \) A. Determine if the system containing \( u, v, \) and \( b \) is consistent. If the system is consistent, \( b \) might be in Span \(\{u,v\}\).
- \( \quad \) B. Determine if the system containing \( u, v, \) and \( b \) is consistent. If the system is consistent, then \( b \) is not in Span \(\{u,v\}\).
- \( \quad \) C. Determine if the system containing \( u, v, \) and \( b \) is consistent. If the system is consistent, then \( b \) is in Span \(\{u,v\}\).
- \( \quad \) D. Determine if the system containing \( u, v, \) and \( b \) is consistent. If the system is inconsistent, then \( b \) is in Span \(\{u,v\}\).
**Let \( b = \begin{bmatrix} h \\ k \end{bmatrix} \). Find the augmented matrix \([u \, v \, b]\).**
\[ \begin{bmatrix} 11 & 11 & h \\ -1 & 1 & k \end{bmatrix} \]
**How is a system determined as consistent?**
- \( \quad \) A. A system is consistent if there is one solution or infinitely many solutions.
- \( \quad \) B. A system is consistent only if all of the variables equal each other.
- \( \quad \) C. A system is consistent if there are no solutions.
- \( \quad \) D. Solve for the variables after setting the equations equal to 0.
**Row reduce the augmented matrix to its reduced echelon form.**
\[](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc0867650-b445-489d-98f3-e27e64467d14%2Fd250f33a-b08a-4655-9489-c70032fde7c1%2Fpiw0shv_processed.png&w=3840&q=75)
Transcribed Image Text:**Linear Algebra: Span and Consistency**
Let \( u = \begin{bmatrix} 11 \\ -1 \end{bmatrix} \) and \( v = \begin{bmatrix} 11 \\ 1 \end{bmatrix} \). Show that \( \begin{bmatrix} h \\ k \end{bmatrix} \) is in Span \(\{u,v\}\) for all \( h \) and \( k \).
**How can it be shown that a vector \( b \) is in Span \(\{u, v\}\)?**
- \( \quad \) A. Determine if the system containing \( u, v, \) and \( b \) is consistent. If the system is consistent, \( b \) might be in Span \(\{u,v\}\).
- \( \quad \) B. Determine if the system containing \( u, v, \) and \( b \) is consistent. If the system is consistent, then \( b \) is not in Span \(\{u,v\}\).
- \( \quad \) C. Determine if the system containing \( u, v, \) and \( b \) is consistent. If the system is consistent, then \( b \) is in Span \(\{u,v\}\).
- \( \quad \) D. Determine if the system containing \( u, v, \) and \( b \) is consistent. If the system is inconsistent, then \( b \) is in Span \(\{u,v\}\).
**Let \( b = \begin{bmatrix} h \\ k \end{bmatrix} \). Find the augmented matrix \([u \, v \, b]\).**
\[ \begin{bmatrix} 11 & 11 & h \\ -1 & 1 & k \end{bmatrix} \]
**How is a system determined as consistent?**
- \( \quad \) A. A system is consistent if there is one solution or infinitely many solutions.
- \( \quad \) B. A system is consistent only if all of the variables equal each other.
- \( \quad \) C. A system is consistent if there are no solutions.
- \( \quad \) D. Solve for the variables after setting the equations equal to 0.
**Row reduce the augmented matrix to its reduced echelon form.**
\[
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