Problem 31 - Advanced Engineering Mathematics

Question

In Problems 31 and 32, without solving, state whether the given homogeneous system has only the trivial solution or has infinitely many solutions. $$ \begin{array}{r} x_{1}-x_{2}+x_{3}=0 \\ 5 x_{1}+x_{2}-x_{3}=0 \\ x_{1}+2 x_{2}+x_{3}=0 \end{array} $$

Step-by-Step Solution

Answer

The system has only the trivial solution.

Step 1: Identify the System of Equations

The given system of equations is:\\[\left\{\begin{array}{r} x_{1}-x_{2}+x_{3}=0 \ 5x_{1}+x_{2}-x_{3}=0 \ x_{1}+2x_{2}+x_{3}=0 \end{array} \right.\] This represents a homogeneous system of linear equations.

Step 2: Determine the Number of Equations and Variables

Count the number of variables and equations:- There are 3 variables: \(x_1, x_2, x_3\).- There are 3 equations.

Step 3: Analyze the Homogeneous System

In a homogeneous system, if the number of independent equations is equal to the number of variables, the system has only the trivial solution (i.e., all variables are zero). If there are fewer independent equations than variables, there are infinitely many solutions.

Step 4: Determine Linear Independence of Equations

Check for linear independence: - If any equation is a linear combination of the others, then they are linearly dependent. - Calculate a row-echelon form (or reduced form) for better analysis, but since we're not solving, observe algebraic dependences.

Step 5: Check Coefficients Simply for Dependency

The coefficients can give a hint: - The first and third equations can't be made multiples or direct sums of each other. - The second equation involves significant differences in coefficients compared to the other two.

Step 6: Conclusion: State the Solution Type

Since the system seems to have independent equations comparing coefficients, the independent equations imply that it won't reduce to a set having fewer independent equations than variables.

Key Concepts

Linear Independence

The concept of linear independence is essential in determining the nature of solutions in a system of equations. Linear independence refers to a situation where no equation in a system can be written as a linear combination of the others. This means that each equation provides unique information about the variables involved. In the context of a homogeneous system like the one we're analyzing, the independence of the equations is significant. With three equations involving three variables, if all the equations are linearly independent, each equation contributes to narrowing down the possible solutions. This helps ensure that the only solution is the trivial one, where all variables are equal to zero. To check for linear independence without solving, observe if:

• Any equation is a multiple of another.
• Any equation is a sum or difference of other equations.

If neither condition is met, the system has no redundancy among equations, signifying linear independence.

Trivial Solution

A trivial solution in the context of homogeneous systems occurs when the only solution is the one where all the variables are zero. This represents a situation where the system doesn't generate any new information beyond what's evident directly from equations' structure. It’s the simplest solution form possible, found by setting all variables to zero and checking if they satisfy every equation in the system. For our given system:

• When each equation is set with the variables as zero,
• The conditions are naturally satisfied:
  • \( x_1 - x_2 + x_3 = 0 \) translates naturally to \( 0 = 0 \), as do the others.

Thus, when the system is determined to have independent equations equal in number to variables, it confirms that the trivial solution is the sole solution.

Infinitely Many Solutions

In contrast to a system with only a trivial solution, a system with infinitely many solutions is one where there are fewer independent equations than variables. This situation arises when some equations in the system are dependent upon others. Dependency can be observed if:

• One equation can be expressed as a combination of the others.
• There is a redundant equation that doesn't add new constraints to the solution set.

In such cases, there is more freedom in choosing values for the variables, leading to multiple solutions. Since this is not the case for our homogeneous system, due to the linear independence of the equations, infinitely many solutions do not exist here. When analyzing systems, recognizing the balance or imbalance between equations and variables helps in identifying whether solutions extend beyond the trivial.