Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![### Educational Content: Solution of Ax = 0 in Parametric Vector Form
#### Problem Statement
Describe all solutions of \( Ax = 0 \) in parametric vector form, where \( A \) is row equivalent to the given matrix:
\[
\begin{bmatrix}
1 & 3 & -4 & 7 \\
0 & 1 & -2 & 4
\end{bmatrix}
\]
#### Parametric Vector Form
The solution \( x \) is expressed as:
\[
x = x_3
\begin{bmatrix}
\_ \\
\_ \\
\_ \\
\_
\end{bmatrix}
+ x_4
\begin{bmatrix}
\_ \\
\_ \\
\_ \\
\_
\end{bmatrix}
\]
**Note**: The blanks in the vector representation should be filled with integers or fractions derived from solving the equations generated by the matrix.
#### Instructions
To complete the solution, substitute the appropriate values for each component of the vectors by solving the system of equations denoted by the matrix.
This activity involves:
- Identifying free variables (typically \( x_3 \) and \( x_4 \) in this case).
- Expressing each dependent variable in terms of the free variables.
- Writing down the general solution in parametric vector form.
#### Concept Explanation
In this exercise, we are dealing with a system of linear equations represented in matrix form. The goal is to find all possible solutions, expressed in a concise vector form, of the homogeneous equation \( Ax = 0 \). Here, \( A \) has been row-reduced, simplifying the process of finding these solutions.
This method is integral in linear algebra, allowing for a comprehensive understanding of solution spaces and the behavior of linear transformations.](https://content.bartleby.com/qna-images/question/704a64b5-5250-41d0-9c29-5aaf5a50e535/badcea68-e50f-44e4-96fd-413afabd5a6b/gk35uu8_processed.jpeg)
Transcribed Image Text:### Educational Content: Solution of Ax = 0 in Parametric Vector Form
#### Problem Statement
Describe all solutions of \( Ax = 0 \) in parametric vector form, where \( A \) is row equivalent to the given matrix:
\[
\begin{bmatrix}
1 & 3 & -4 & 7 \\
0 & 1 & -2 & 4
\end{bmatrix}
\]
#### Parametric Vector Form
The solution \( x \) is expressed as:
\[
x = x_3
\begin{bmatrix}
\_ \\
\_ \\
\_ \\
\_
\end{bmatrix}
+ x_4
\begin{bmatrix}
\_ \\
\_ \\
\_ \\
\_
\end{bmatrix}
\]
**Note**: The blanks in the vector representation should be filled with integers or fractions derived from solving the equations generated by the matrix.
#### Instructions
To complete the solution, substitute the appropriate values for each component of the vectors by solving the system of equations denoted by the matrix.
This activity involves:
- Identifying free variables (typically \( x_3 \) and \( x_4 \) in this case).
- Expressing each dependent variable in terms of the free variables.
- Writing down the general solution in parametric vector form.
#### Concept Explanation
In this exercise, we are dealing with a system of linear equations represented in matrix form. The goal is to find all possible solutions, expressed in a concise vector form, of the homogeneous equation \( Ax = 0 \). Here, \( A \) has been row-reduced, simplifying the process of finding these solutions.
This method is integral in linear algebra, allowing for a comprehensive understanding of solution spaces and the behavior of linear transformations.
Expert Solution
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Step 1
Consider the given matrix as,
Now this matrix can be write in the form of
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