Find a basis of the following linear system. { X1 2x1 x1 3x3 4x2 x2 + x3 + 2x2 + + 3x3 + 7x4 7x4 11x4 0 0 = 0 = =
Find a basis of the following linear system. { X1 2x1 x1 3x3 4x2 x2 + x3 + 2x2 + + 3x3 + 7x4 7x4 11x4 0 0 = 0 = =
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
![**Problem Statement:**
Find a basis of the following linear system:
\[
\begin{align*}
x_1 - 4x_2 - 3x_3 - 7x_4 &= 0 \\
2x_1 - x_2 + x_3 + 7x_4 &= 0 \\
x_1 + 2x_2 + 3x_3 + 11x_4 &= 0 \\
\end{align*}
\]
**Objective:**
Identify a set of linearly independent vectors that span the solution space of the given homogeneous system of linear equations.
**Explanation of the System:**
The system consists of three linear equations with four unknowns \(x_1, x_2, x_3,\) and \(x_4\). The goal is to determine a basis for the solutions, which involves finding vectors for which any solution to the system can be expressed as a linear combination.
**Steps to Solve:**
1. **Form the Augmented Matrix**: Construct the matrix representing the coefficients of the variables stored in order.
2. **Row Reduce to Echelon Form**: Use row operations to bring the matrix to row-echelon form or reduced row-echelon form.
3. **Identify Free Variables**: Determine which variables are leading ones in each row and which are free.
4. **Express Solutions**: Write the solutions in terms of the free variables. Each free variable can correspond to a basis vector.
5. **Determine Basis Vectors**: From the parametric form of the general solution, extract basis vectors.
**Conclusion:**
This process will lead to identifying a set of vectors that form a basis for the solution space, capturing all possible solutions of the given linear system. The number of basis vectors will indicate the dimension of the solution space.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc303f822-5e20-4d30-8920-178170554c96%2F3c2e14fd-2bda-4123-b559-a302a13fe341%2F402demqf_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find a basis of the following linear system:
\[
\begin{align*}
x_1 - 4x_2 - 3x_3 - 7x_4 &= 0 \\
2x_1 - x_2 + x_3 + 7x_4 &= 0 \\
x_1 + 2x_2 + 3x_3 + 11x_4 &= 0 \\
\end{align*}
\]
**Objective:**
Identify a set of linearly independent vectors that span the solution space of the given homogeneous system of linear equations.
**Explanation of the System:**
The system consists of three linear equations with four unknowns \(x_1, x_2, x_3,\) and \(x_4\). The goal is to determine a basis for the solutions, which involves finding vectors for which any solution to the system can be expressed as a linear combination.
**Steps to Solve:**
1. **Form the Augmented Matrix**: Construct the matrix representing the coefficients of the variables stored in order.
2. **Row Reduce to Echelon Form**: Use row operations to bring the matrix to row-echelon form or reduced row-echelon form.
3. **Identify Free Variables**: Determine which variables are leading ones in each row and which are free.
4. **Express Solutions**: Write the solutions in terms of the free variables. Each free variable can correspond to a basis vector.
5. **Determine Basis Vectors**: From the parametric form of the general solution, extract basis vectors.
**Conclusion:**
This process will lead to identifying a set of vectors that form a basis for the solution space, capturing all possible solutions of the given linear system. The number of basis vectors will indicate the dimension of the solution space.
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