Suppose the solution set of a certain system of linear equations can be described as x₁ = 7+ 4x3, x2 = -2-7x3, with x3 free. Use vectors to describe this set as a line in R³. Geometrically, the solution set is a line through 7 -2 0 parallel to 4 -7 1
Suppose the solution set of a certain system of linear equations can be described as x₁ = 7+ 4x3, x2 = -2-7x3, with x3 free. Use vectors to describe this set as a line in R³. Geometrically, the solution set is a line through 7 -2 0 parallel to 4 -7 1
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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![**Understanding Vector Representation in Linear Equations**
Suppose the solution set of a certain system of linear equations can be described as \( x_1 = 7 + 4x_3 \) and \( x_2 = -2 - 7x_3 \), with \( x_3 \) being free. Use vectors to describe this set as a line in \( \mathbb{R}^3 \).
---
Geometrically, the solution set is a line through:
\[
\begin{bmatrix}
7 \\
-2 \\
0
\end{bmatrix}
\]
parallel to:
\[
\begin{bmatrix}
4 \\
-7 \\
1
\end{bmatrix}
\]
---
In this explanation, \( x_3 \) is a free variable, implying it can take any real value. As \( x_3 \) varies, the coordinates \( x_1 \) and \( x_2 \) change accordingly, tracing out a line in \( \mathbb{R}^3 \). The first vector \(\begin{bmatrix} 7 \\ -2 \\ 0 \end{bmatrix}\) represents a specific point through which this line passes, while the second vector \(\begin{bmatrix} 4 \\ -7 \\ 1 \end{bmatrix}\) indicates the direction of the line, showing how the coordinates change as \( x_3 \) varies.
### Interpretation:
- **Starting Point**: \(\begin{bmatrix} 7 \\ -2 \\ 0 \end{bmatrix}\)
- **Direction of the Line**: \(\begin{bmatrix} 4 \\ -7 \\ 1 \end{bmatrix}\)
Any point on the line can be expressed as a combination of the starting point and a scalar multiple of the direction vector.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8b6c3689-2b60-4c25-801e-fccc9acdd031%2Fa49b05b8-b8de-463a-9021-a782f2d0554a%2Fin3yka_processed.png&w=3840&q=75)
Transcribed Image Text:**Understanding Vector Representation in Linear Equations**
Suppose the solution set of a certain system of linear equations can be described as \( x_1 = 7 + 4x_3 \) and \( x_2 = -2 - 7x_3 \), with \( x_3 \) being free. Use vectors to describe this set as a line in \( \mathbb{R}^3 \).
---
Geometrically, the solution set is a line through:
\[
\begin{bmatrix}
7 \\
-2 \\
0
\end{bmatrix}
\]
parallel to:
\[
\begin{bmatrix}
4 \\
-7 \\
1
\end{bmatrix}
\]
---
In this explanation, \( x_3 \) is a free variable, implying it can take any real value. As \( x_3 \) varies, the coordinates \( x_1 \) and \( x_2 \) change accordingly, tracing out a line in \( \mathbb{R}^3 \). The first vector \(\begin{bmatrix} 7 \\ -2 \\ 0 \end{bmatrix}\) represents a specific point through which this line passes, while the second vector \(\begin{bmatrix} 4 \\ -7 \\ 1 \end{bmatrix}\) indicates the direction of the line, showing how the coordinates change as \( x_3 \) varies.
### Interpretation:
- **Starting Point**: \(\begin{bmatrix} 7 \\ -2 \\ 0 \end{bmatrix}\)
- **Direction of the Line**: \(\begin{bmatrix} 4 \\ -7 \\ 1 \end{bmatrix}\)
Any point on the line can be expressed as a combination of the starting point and a scalar multiple of the direction vector.
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