Question

Consider the following statements about a system of linear equations with augmented matrix \(A\). In each case either prove the statement or give an example for which it is false. a. If the system is homogeneous, every solution is trivial. b. If the system has a nontrivial solution, it cannot be homogeneous. c. If there exists a trivial solution, the system is homogeneous. d. If the system is consistent, it must be homogeneous. Now assume that the system is homogeneous. e. If there exists a nontrivial solution, there is no trivial solution. f. If there exists a solution, there are infinitely many solutions. g. If there exist nontrivial solutions, the row-echelon form of \(A\) has a row of zeros. h. If the row-echelon form of \(A\) has a row of zeros, there exist nontrivial solutions. i. If a row operation is applied to the system, the new system is also homogeneous.

Short answer

a: False; b: False; c: False; d: False; e: False; f: True; g: True; h: True; i: True.

Step-by-step solution

Analyze Claim 'a'

The statement claims that if a system is homogeneous, every solution is trivial. In a homogeneous system, the equation can be represented as \( Ax = 0 \), where \( A \) is the coefficient matrix. The zero vector is always a solution to homogeneous systems, also known as the trivial solution. However, if the system is singular (i.e. \( ext{rank}(A) < ext{number of unknowns} \)), there will also be nontrivial (non-zero) solutions. Therefore, the claim is false; a homogeneous system always has the trivial solution, but it may also have nontrivial solutions.

Analyze Claim 'b'

The statement claims that if the system has a nontrivial solution, it cannot be homogeneous. If a system is homogeneous, it is given by \( Ax = 0 \). Homogeneous systems can have both trivial and nontrivial solutions, particularly when the matrix \( A \) is singular. Thus, a homogeneous system can have nontrivial solutions, making this claim false.

Analyze Claim 'c'

The statement claims that if there exists a trivial solution, the system is homogeneous. A trivial solution occurs in homogeneous systems because the zero vector satisfies \( Ax = 0 \). However, the existence of a trivial solution does not necessarily mean a system is homogeneous, as any system with a zero output (e.g., a zero free term in inhomogeneous systems) may also present the zero vector as a solution. Thus, the claim is false.

Analyze Claim 'd'

The statement claims that if a system is consistent, it must be homogeneous. A consistent system simply means there exists at least one solution. Non-homogeneous systems (e.g., those of the form \( Ax = b \) with \( b eq 0 \)) can also be consistent. Therefore, consistency does not imply homogeneity; hence this statement is false.

Analyze Claim 'e'

Given that the system is homogeneous, if there exists a nontrivial solution, there is no trivial solution. This is false because in any homogeneous system, the trivial solution (zero vector) always exists, regardless of the existence of any nontrivial solutions.

Analyze Claim 'f'

The statement implies that if there exists a solution to a homogeneous system, there are infinitely many solutions. For a homogeneous system \( Ax = 0 \), having a nontrivial solution implies that the null space has more than just the zero solution, which means the null space is a nontrivial subspace. This suggests infinitely many solutions. Thus, this claim is true.

Analyze Claim 'g'

The statement is that if there exist nontrivial solutions, the row-echelon form of \( A \) has a row of zeros. Nontrivial solutions imply a linearly dependent system, which also suggests that the rank of \( A \) is less than the number of unknowns. Therefore, there must be free variables indicating a row of zeros in the row-echelon form. Hence, this statement is true.

Analyze Claim 'h'

The statement claims that if the row-echelon form of \( A \) has a row of zeros, there exist nontrivial solutions. This implies there are more variables than the rank (the system is under-determined), leading to free variables. Hence, the existence of nontrivial solutions is ensured, making the statement true.

Analyze Claim 'i'

The statement claims that if a row operation is applied to the system, the new system is also homogeneous. Row operations do not change the fundamental nature of a system being homogeneous, since they preserve the zero vector solution. Therefore, this statement is true.

Key concepts

Homogeneous Systems

A homogeneous system of linear equations is one where all of the equations set equal to zero. It can be represented as \( Ax = 0 \), where \( A \) is the coefficient matrix and \( x \) is a vector of variables. In this system, the trivial solution, which is the zero vector \( x = 0 \), always exists. Yet, homogeneous systems are interesting because they can also have nontrivial solutions. These occur when the matrix \( A \) is not full rank, meaning the number of leading ones in reduced row-echelon form is less than the number of variables. This condition makes the system singular, leading to the existence of nontrivial solutions, possibly forming infinite solution sets. Understanding homogeneous systems is crucial as they form the foundation for understanding vector spaces and linear transformations.

Trivial and Nontrivial Solutions

In the context of homogeneous systems, solutions are classified as trivial or nontrivial. A trivial solution is simply the zero vector; all variables are set to zero, which satisfies the equation \( Ax = 0 \). This solution is always present because substituting zero for any variables results in zero, maintaining the equality.
Nontrivial solutions, on the other hand, are solutions where not all variables are zero. These solutions exist when the system matrix \( A \) is singular, meaning it has linearly dependent rows. In this scenario, the solution space forms a subspace that is more complex than just the trivial solution, often containing infinitely many solutions. Nontrivial solutions are vital for applications where simple zero outcomes aren't meaningful, such as in engineering and physics.

Row-Echelon Form

Row-echelon form is a way of organizing a matrix to simplify solving a system of linear equations. It involves using row operations to arrange the matrix such that beneath each leading entry in a row, all entries are zeroes. This form helps identify the pivot variables and simplifies solving equations, particularly useful in determining the rank of the matrix.
In a homogeneous system, if the row-echelon form has a row of zeros, it indicates the presence of free variables, which are not leading. Free variables mean there are more variables than independent equations, suggesting nontrivial solutions exist. Such a structure implies that the solution set is a line, plane, or higher-dimensional space in which infinite solutions can lie, emphasizing its utility in linear algebra.

Consistency of Systems

Consistency of a system of linear equations refers to whether there exists at least one solution. A system is consistent if there is a solution; otherwise, it is inconsistent. Homogeneous systems are inherently consistent because the trivial solution \( x = 0 \) always satisfies \( Ax = 0 \). However, non-homogeneous systems, where equations are of the form \( Ax = b \) with \( b eq 0 \), can also be consistent if there's a specific solution satisfying the equations.
It is important to check for consistency when solving linear equations to ensure valid solutions exist. This involves evaluating the augmented matrix and determining if the rank condition holds: that the rank of the augmented matrix equals the rank of the coefficient matrix. Understanding consistency helps avoid futile attempts to solve systems with no real solutions.