Let a₁ = 1 5 -1 a₂ -6 - 25 3 and b = 3 -5 . For what value(s) of h is b in the plane spanned by a, and a₂? h The value(s) of h is(are). (Use a comma to separate answers as needed.)
Let a₁ = 1 5 -1 a₂ -6 - 25 3 and b = 3 -5 . For what value(s) of h is b in the plane spanned by a, and a₂? h The value(s) of h is(are). (Use a comma to separate answers as needed.)
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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PLEASE SOLVE BOTH I WILL GIVE THUMBS UP!!!!
![**Linear Algebra Problem: Finding the Value of h**
Consider the following vectors defined in \(\mathbb{R}^3\):
\[
\mathbf{a_1} = \begin{bmatrix} 1 \\ 5 \\ -1 \end{bmatrix},\ \mathbf{a_2} = \begin{bmatrix} -6 \\ -25 \\ 3 \end{bmatrix}, \ \text{and} \ \mathbf{b} = \begin{bmatrix} 3 \\ -5 \\ h \end{bmatrix}
\]
**Question:**
For what value(s) of \(h\) is \(\mathbf{b}\) in the plane spanned by \(\mathbf{a_1}\) and \(\mathbf{a_2}\)?
To determine this, we must check if \(\mathbf{b}\) lies in the span of \(\mathbf{a_1}\) and \(\mathbf{a_2}\). This can be expressed as whether \(\mathbf{b}\) can be written as a linear combination of \(\mathbf{a_1}\) and \(\mathbf{a_2}\):
\[
c_1 \mathbf{a_1} + c_2 \mathbf{a_2} = \mathbf{b}
\]
This expands to the following system of linear equations:
\[
c_1 \begin{bmatrix} 1 \\ 5 \\ -1 \end{bmatrix} + c_2 \begin{bmatrix} -6 \\ -25 \\ 3 \end{bmatrix} = \begin{bmatrix} 3 \\ -5 \\ h \end{bmatrix}
\]
**Solution:**
Solve for \(c_1\), \(c_2\), and \(h\).
The value(s) of \(h\) is(are) \(\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\). (Use a comma to separate answers as needed.)
[Solution Steps for the Instructor to Include]:
- Set up the augmented matrix from the system of equations.
- Row-reduce the matrix to determine the consistency of the system with respect to \(h\).
- Solve for \(h\) where appropriate, ensuring the solution vector satisfies the original system.
This exercise aims to understand vector spaces, linear combinations, and the geometric](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1c3522a4-50b1-4726-a83e-d332776e45d2%2Fd3932981-f979-45b4-b6c1-826462f24429%2Fdzylod5_processed.png&w=3840&q=75)
Transcribed Image Text:**Linear Algebra Problem: Finding the Value of h**
Consider the following vectors defined in \(\mathbb{R}^3\):
\[
\mathbf{a_1} = \begin{bmatrix} 1 \\ 5 \\ -1 \end{bmatrix},\ \mathbf{a_2} = \begin{bmatrix} -6 \\ -25 \\ 3 \end{bmatrix}, \ \text{and} \ \mathbf{b} = \begin{bmatrix} 3 \\ -5 \\ h \end{bmatrix}
\]
**Question:**
For what value(s) of \(h\) is \(\mathbf{b}\) in the plane spanned by \(\mathbf{a_1}\) and \(\mathbf{a_2}\)?
To determine this, we must check if \(\mathbf{b}\) lies in the span of \(\mathbf{a_1}\) and \(\mathbf{a_2}\). This can be expressed as whether \(\mathbf{b}\) can be written as a linear combination of \(\mathbf{a_1}\) and \(\mathbf{a_2}\):
\[
c_1 \mathbf{a_1} + c_2 \mathbf{a_2} = \mathbf{b}
\]
This expands to the following system of linear equations:
\[
c_1 \begin{bmatrix} 1 \\ 5 \\ -1 \end{bmatrix} + c_2 \begin{bmatrix} -6 \\ -25 \\ 3 \end{bmatrix} = \begin{bmatrix} 3 \\ -5 \\ h \end{bmatrix}
\]
**Solution:**
Solve for \(c_1\), \(c_2\), and \(h\).
The value(s) of \(h\) is(are) \(\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\). (Use a comma to separate answers as needed.)
[Solution Steps for the Instructor to Include]:
- Set up the augmented matrix from the system of equations.
- Row-reduce the matrix to determine the consistency of the system with respect to \(h\).
- Solve for \(h\) where appropriate, ensuring the solution vector satisfies the original system.
This exercise aims to understand vector spaces, linear combinations, and the geometric
![The image shows a mathematical problem centered around vector spaces and linear algebra. Below is the transcription of the content suitable for an educational website.
---
**Linear Algebra Problem: Vector Spaces and Span**
Given the matrix \( A \) and vector \( b \):
\[
A = \begin{pmatrix}
1 & 0 & -6 \\
0 & 3 & -5 \\
-5 & 9 & 4
\end{pmatrix}
\]
\[
b = \begin{pmatrix}
9 \\
-2 \\
-29
\end{pmatrix}
\]
Denote the columns of \( A \) by \( \mathbf{a_1}, \mathbf{a_2}, \mathbf{a_3} \), and let \( W = \text{Span} \{ \mathbf{a_1}, \mathbf{a_2}, \mathbf{a_3} \} \).
**Questions:**
a. Is \( \mathbf{b} \) in \( \{ \mathbf{a_1}, \mathbf{a_2}, \mathbf{a_3} \} \)? How many vectors are in \( \{ \mathbf{a_1}, \mathbf{a_2}, \mathbf{a_3} \} \)?
b. Is \( \mathbf{b} \) in \( W \)? How many vectors are in \( W \)?
c. Show that \( \mathbf{a_2} \) is in \( W \). [Hint: Row operations are unnecessary.]
---
**Step-by-Step Solution:**
a. **Is \( \mathbf{b} \) in \( \{ \mathbf{a_1}, \mathbf{a_2}, \mathbf{a_3} \} \)? Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.**
- A. \(\bigcirc\) No, \( \mathbf{b} \) is not in \( \{ \mathbf{a_1}, \mathbf{a_2}, \mathbf{a_3} \} \) since \( \mathbf{b} \) is not equal to \( \mathbf{a_1}, \mathbf{a_2}, \) or \( \mathbf{a_3} \).
- B. \(\bigcirc\) Yes,](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1c3522a4-50b1-4726-a83e-d332776e45d2%2Fd3932981-f979-45b4-b6c1-826462f24429%2F74srq4p_processed.png&w=3840&q=75)
Transcribed Image Text:The image shows a mathematical problem centered around vector spaces and linear algebra. Below is the transcription of the content suitable for an educational website.
---
**Linear Algebra Problem: Vector Spaces and Span**
Given the matrix \( A \) and vector \( b \):
\[
A = \begin{pmatrix}
1 & 0 & -6 \\
0 & 3 & -5 \\
-5 & 9 & 4
\end{pmatrix}
\]
\[
b = \begin{pmatrix}
9 \\
-2 \\
-29
\end{pmatrix}
\]
Denote the columns of \( A \) by \( \mathbf{a_1}, \mathbf{a_2}, \mathbf{a_3} \), and let \( W = \text{Span} \{ \mathbf{a_1}, \mathbf{a_2}, \mathbf{a_3} \} \).
**Questions:**
a. Is \( \mathbf{b} \) in \( \{ \mathbf{a_1}, \mathbf{a_2}, \mathbf{a_3} \} \)? How many vectors are in \( \{ \mathbf{a_1}, \mathbf{a_2}, \mathbf{a_3} \} \)?
b. Is \( \mathbf{b} \) in \( W \)? How many vectors are in \( W \)?
c. Show that \( \mathbf{a_2} \) is in \( W \). [Hint: Row operations are unnecessary.]
---
**Step-by-Step Solution:**
a. **Is \( \mathbf{b} \) in \( \{ \mathbf{a_1}, \mathbf{a_2}, \mathbf{a_3} \} \)? Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.**
- A. \(\bigcirc\) No, \( \mathbf{b} \) is not in \( \{ \mathbf{a_1}, \mathbf{a_2}, \mathbf{a_3} \} \) since \( \mathbf{b} \) is not equal to \( \mathbf{a_1}, \mathbf{a_2}, \) or \( \mathbf{a_3} \).
- B. \(\bigcirc\) Yes,
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