91Ó°ÊÓ

Determine whether the solution space of the system \(A \mathbf{x}=0\) is a line through the origin, a plane through the origin, or the origin only. If it is a plane, find an equation for it. If it is a line, find parametric equations for it. \(A=\left[\begin{array}{rrr}-1 & 1 & 1 \\ 3 & -1 & 0 \\ 2 & -4 & -5\end{array}\right]\) \(A=\left[\begin{array}{rrr}1 & -2 & 3 \\ -3 & 6 & 9 \\ -2 & 4 & -6\end{array}\right]\) \(A=\left[\begin{array}{lll}1 & 2 & 3 \\ 2 & 5 & 3 \\ 1 & 0 & 8\end{array}\right]\) \(A=\left[\begin{array}{rrr}1 & 2 & -6 \\ 1 & 4 & 4 \\ 3 & 10 & 6\end{array}\right]\) \(A=\left[\begin{array}{rrr}1 & -1 & 1 \\ 2 & -1 & 4 \\ 3 & 1 & 11\end{array}\right]\) \(A=\left[\begin{array}{lll}1 & -3 & 1 \\ 2 & -6 & 2 \\ 3 & -9 & 3\end{array}\right]\)

Step by step solution

01

Analyze matrix rank for each system

The solution space of the system \( A \mathbf{x} = 0 \) is determined by the rank of matrix \( A \). If the rank \( r \) is 3, the solution is the origin only. If \( r = 2 \), it is a line through the origin. If \( r = 1 \), it is a plane through the origin.

02

Compute rank for \( A = \left[ \begin{array}{rrr}-1 & 1 & 1 \\ 3 & -1 & 0 \\ 2 & -4 & -5\end{array}\right] \)

After row operations, the matrix reduces to \( \begin{bmatrix} 1 & -1 & -1 \ 0 & 1 & 3 \ 0 & 0 & 0\end{bmatrix} \). The rank is 2, meaning the solution space is a line.

03

Parametric form for line (First matrix)

To find the parametric equations, set the free variable, say \( x_3 = t \), and express \( x_1, x_2 \) in terms of \( t \). Solving, \( x_1 = -2t \), \( x_2 = -3t \), and \( x_3 = t \). So the parametric equations are \( x_1 = -2t \), \( x_2 = -3t \), \( x_3 = t \).

04

Compute rank for \( A = \left[\begin{array}{rrr}1 & -2 & 3 \\ -3 & 6 & 9 \\ -2 & 4 & -6\end{array}\right] \)

After row operations, the matrix reduces to \( \begin{bmatrix} 1 & -2 & 3 \ 0 & 0 & 0 \ 0 & 0 & 0\end{bmatrix} \). The rank is 1, meaning the solution space is a plane.

05

Equation of the plane (Second matrix)

The equation that relates the variables in a plane is \( 1x_1 - 2x_2 + 3x_3 = 0 \).

06

Compute rank for \( A = \left[\begin{array}{lll}1 & 2 & 3 \\ 2 & 5 & 3 \\ 1 & 0 & 8\end{array}\right] \)

After row operations, the matrix reduces to \( \begin{bmatrix} 1 & 2 & 3 \ 0 & 1 & -3 \ 0 & 0 & 4\end{bmatrix} \). The rank is 3, meaning the solution space is the origin only.

07

Compute rank for \( A = \left[ \begin{array}{rrr} 1 & 2 & -6 \\ 1 & 4 & 4 \\ 3 & 10 & 6\end{array}\right] \)

After row operations, the matrix reduces to \( \begin{bmatrix} 1 & 2 & -6 \ 0 & 2 & 10 \ 0 & 0 & 0\end{bmatrix} \). The rank is 2, meaning the solution space is a line.

08

Parametric form for line (Fourth matrix)

Set one of the variables as a free variable, e.g., \( x_3 = t \). Solving for \( x_1 \) and \( x_2 \), give \( x_1 = t \), \( x_2 = -5t \), \( x_3 = t \). So the parametric representation is \( x_1 = t \), \( x_2 = -5t \), \( x_3 = t \).

09

Compute rank for \( A = \left[\begin{array}{rrr}1 & -1 & 1 \\ 2 & -1 & 4 \\ 3 & 1 & 11\end{array}\right] \)

After row operations, the matrix reduces to \( \begin{bmatrix} 1 & -1 & 1 \ 0 & 1 & -3 \ 0 & 0 & 0\end{bmatrix} \). The rank is 2, meaning the solution space is a line.

10

Parametric form for line (Fifth matrix)

Choose a free variable, \( x_3 = t \), then \( x_1 = 2t \), \( x_2 = 3t \), \( x_3 = t \). Parametric equations are \( x_1 = 2t \), \( x_2 = 3t \), \( x_3 = t \).

11

Compute rank for \( A =\left[\begin{array}{lll}1 & -3 & 1 \\ 2 & -6 & 2 \\ 3 & -9 & 3\end{array}\right] \)

Each row is a scalar multiple of the others. The matrix can be reduced to \( \begin{bmatrix} 1 & -3 & 1 \ 0 & 0 & 0 \ 0 & 0 & 0\end{bmatrix} \). The rank is 1, meaning the solution space is a plane.

12

Equation of the plane (Sixth matrix)

The equation reflecting the solution space is \( 1x_1 - 3x_2 + 1x_3 = 0 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Matrix Rank

The concept of matrix rank is central to understanding the nature of the solution space in linear systems. The rank of a matrix essentially tells us the number of independent rows or columns in the matrix. For linear systems of equations, particularly those in the form of matrix equations like \( A \mathbf{x} = 0 \), the rank provides insights into the dimensionality of the solution space.

• If the rank of matrix \( A \) is equal to the number of variables, the only solution is the trivial solution, implying the solution space is just the origin.
• If the rank is one less than the number of variables, the solution space forms a line through the origin. The deficit in the rank indicates that there is one free variable, which determines the line.
• If the rank is two less than the number of variables, the solution space becomes a plane through the origin. This means there are two free variables to describe all the solution points.

Understanding and computing the rank is foundational before solving any system of equations. Identifying the correct rank allows us to determine the nature of the solution space.

Parametric Equations

In the context of linear equations, parametric equations are a way of expressing the coordinates of all the solutions of a system using one or more parameters. When we have a free variable due to a lower matrix rank, this leads to a solution space, typically in the form of a line or plane.

For instance, consider a system resulting in a line as the solution space. If rank analysis shows that the rank is two (in a system of three variables), a free variable, say \( t \), can be introduced. The other variables are defined in terms of \( t \).

• The parametric equations capture the entire set of solutions by varying \( t \).
• For a line through points \(x_1 = at + b \, x_2 = ct + d \, x_3 = et + f\), the parameters \( a, c, e \) describe the direction of the line.

Parametric equations are powerful for visualization and understanding the structure of the solution set.

Linear Equations

Linear equations are the building blocks of algebra and intersection points for various algebraic concepts. These equations take the form \( ax + by + cz = d \), where \( a, b, c \) are coefficients and \( d \) is a constant. A crucial element in understanding these equations lies in their matrix representation, which succinctly encapsulates multiple equations.

In systems like \( A \mathbf{x} = 0 \), there is often no constant term; the system is homogeneous, meaning every solution passes through the origin. The task is to determine the solution space configuration: whether it is a point, a line, or a plane.

• The simplicity of the equations makes them suitable for using matrix operations to find solutions.
• Solutions for these types of equations typically require techniques like row reduction to determine the rank first.

Grasping the basic ideas about linear equations prepares students for handling more complex algebraic and geometric problems.

Homogeneous Systems

A homogeneous system is a system of linear equations in which all of the constant terms are zero. This is represented in the form \( A \mathbf{x} = 0 \). What makes homogeneous systems interesting is that they always have at least one solution—the trivial solution, where all variables are zero.

• Understanding homogeneous systems is critical for grasping why solution spaces are structured as they are.
• Their solution space either touches the origin or entirely consists of it, as determined by the matrix rank.
• They provide insights into the vector spaces as each solution can be seen as a vector, with these vectors forming a span in a space of certain dimensionality.

Solving these systems is crucial in many areas of applied mathematics, physics, and engineering, where they model a variety of equilibrium and steady-state scenarios.