4 V = V2 = y = For what value(s) of h is vector y in the plane generated by v1, V2, V3? Show all your work, do not skip steps. Displaying only the final answer is not enough to get credit.
4 V = V2 = y = For what value(s) of h is vector y in the plane generated by v1, V2, V3? Show all your work, do not skip steps. Displaying only the final answer is not enough to get credit.
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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![### Linear Algebra Problem: Determining Values for Vector Coefficients
Consider the vectors defined as follows:
\[
\mathbf{v_1} = \begin{pmatrix}
1 \\
0 \\
-1 \\
2
\end{pmatrix} \quad ; \quad \mathbf{v_2} = \begin{pmatrix}
2 \\
3 \\
1 \\
-2
\end{pmatrix} \quad ; \quad \mathbf{v_3} = \begin{pmatrix}
0 \\
0 \\
1 \\
2
\end{pmatrix}
\]
Given the vector:
\[
\mathbf{y} = \begin{pmatrix}
4 \\
3 \\
h \\
0
\end{pmatrix}
\]
Determine the value(s) of \( h \) for which the vector \( \mathbf{y} \) is in the plane generated by the vectors \( \mathbf{v_1}, \mathbf{v_2}, \mathbf{v_3} \).
#### Instructions:
1. Show all your work and do not skip steps.
2. Simply displaying the final answer will not suffice for full credit.
### Solution Outline:
To find the values of \( h \) for which \( \mathbf{y} \) is in the plane generated by \( \mathbf{v_1}, \mathbf{v_2}, \mathbf{v_3} \), we need to express \( \mathbf{y} \) as a linear combination of \( \mathbf{v_1}, \mathbf{v_2}, \mathbf{v_3} \):
\[
\mathbf{y} = c_1 \mathbf{v_1} + c_2 \mathbf{v_2} + c_3 \mathbf{v_3}
\]
This results in the following system of equations:
\[
\begin{cases}
c_1 \cdot 1 + c_2 \cdot 2 + c_3 \cdot 0 = 4 \\
c_1 \cdot 0 + c_2 \cdot 3 + c_3 \cdot 0 = 3 \\
c_1 \cdot (-1) + c_2 \cdot 1 + c_3 \cdot 1 = h \\
c_1 \cd](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdf0a178e-b4ea-4552-8c3f-e1ff6984e724%2Fc61ffc5c-d199-4cda-b267-5bd5230a8af2%2F383hqh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Linear Algebra Problem: Determining Values for Vector Coefficients
Consider the vectors defined as follows:
\[
\mathbf{v_1} = \begin{pmatrix}
1 \\
0 \\
-1 \\
2
\end{pmatrix} \quad ; \quad \mathbf{v_2} = \begin{pmatrix}
2 \\
3 \\
1 \\
-2
\end{pmatrix} \quad ; \quad \mathbf{v_3} = \begin{pmatrix}
0 \\
0 \\
1 \\
2
\end{pmatrix}
\]
Given the vector:
\[
\mathbf{y} = \begin{pmatrix}
4 \\
3 \\
h \\
0
\end{pmatrix}
\]
Determine the value(s) of \( h \) for which the vector \( \mathbf{y} \) is in the plane generated by the vectors \( \mathbf{v_1}, \mathbf{v_2}, \mathbf{v_3} \).
#### Instructions:
1. Show all your work and do not skip steps.
2. Simply displaying the final answer will not suffice for full credit.
### Solution Outline:
To find the values of \( h \) for which \( \mathbf{y} \) is in the plane generated by \( \mathbf{v_1}, \mathbf{v_2}, \mathbf{v_3} \), we need to express \( \mathbf{y} \) as a linear combination of \( \mathbf{v_1}, \mathbf{v_2}, \mathbf{v_3} \):
\[
\mathbf{y} = c_1 \mathbf{v_1} + c_2 \mathbf{v_2} + c_3 \mathbf{v_3}
\]
This results in the following system of equations:
\[
\begin{cases}
c_1 \cdot 1 + c_2 \cdot 2 + c_3 \cdot 0 = 4 \\
c_1 \cdot 0 + c_2 \cdot 3 + c_3 \cdot 0 = 3 \\
c_1 \cdot (-1) + c_2 \cdot 1 + c_3 \cdot 1 = h \\
c_1 \cd
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