Let 7= X1 Describe all solutions of a₁₁ + a₂2 +33 + 44 = Ō. I2 X3 C4 18 = x2 13 ā₂ - [3]₁03 = [27], ₁ = [15] a3 +23 +4
Let 7= X1 Describe all solutions of a₁₁ + a₂2 +33 + 44 = Ō. I2 X3 C4 18 = x2 13 ā₂ - [3]₁03 = [27], ₁ = [15] a3 +23 +4
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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Question
![Let
\[
\vec{a}_1 = \begin{bmatrix} 1 \\ 3 \end{bmatrix}, \,
\vec{a}_2 = \begin{bmatrix} 2 \\ 6 \end{bmatrix}, \,
\vec{a}_3 = \begin{bmatrix} -9 \\ 27 \end{bmatrix}, \,
\vec{a}_4 = \begin{bmatrix} 5 \\ 15 \end{bmatrix}
\]
Describe all solutions of \(\vec{a}_1 x_1 + \vec{a}_2 x_2 + \vec{a}_3 x_3 + \vec{a}_4 x_4 = \vec{0}\).
\[
\vec{x} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix} = x_2
\begin{bmatrix} \, \\ \, \\ \, \\ \, \end{bmatrix} + x_3
\begin{bmatrix} \, \\ \, \\ \, \\ \, \end{bmatrix} + x_4
\begin{bmatrix} \, \\ \, \\ \, \\ \, \end{bmatrix}
\]
This is a linear algebra problem asking to describe all solutions to a linear combination of vectors equaling the zero vector. The vectors \(\vec{a}_1, \vec{a}_2, \vec{a}_3, \vec{a}_4\) are given, and each has a scalar \(x_i\) that represents its contribution to the solution.
The augmented equation and the empty columns suggest constructing a set of solutions based on the free variables \(x_2, x_3, x_4\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F18574973-f25e-4ab8-b7d6-6007b5b87fc4%2Fd2e1470a-459b-4662-a0f8-ed17634f1a1a%2Flzliydi_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let
\[
\vec{a}_1 = \begin{bmatrix} 1 \\ 3 \end{bmatrix}, \,
\vec{a}_2 = \begin{bmatrix} 2 \\ 6 \end{bmatrix}, \,
\vec{a}_3 = \begin{bmatrix} -9 \\ 27 \end{bmatrix}, \,
\vec{a}_4 = \begin{bmatrix} 5 \\ 15 \end{bmatrix}
\]
Describe all solutions of \(\vec{a}_1 x_1 + \vec{a}_2 x_2 + \vec{a}_3 x_3 + \vec{a}_4 x_4 = \vec{0}\).
\[
\vec{x} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix} = x_2
\begin{bmatrix} \, \\ \, \\ \, \\ \, \end{bmatrix} + x_3
\begin{bmatrix} \, \\ \, \\ \, \\ \, \end{bmatrix} + x_4
\begin{bmatrix} \, \\ \, \\ \, \\ \, \end{bmatrix}
\]
This is a linear algebra problem asking to describe all solutions to a linear combination of vectors equaling the zero vector. The vectors \(\vec{a}_1, \vec{a}_2, \vec{a}_3, \vec{a}_4\) are given, and each has a scalar \(x_i\) that represents its contribution to the solution.
The augmented equation and the empty columns suggest constructing a set of solutions based on the free variables \(x_2, x_3, x_4\).
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