Answer:
Refer below.
Explanation:
The statement "The equation Ax = b is not homogeneous if the zero vector is a solution to that system" requires careful examination. Let's analyze the concepts of homogeneous and non-homogeneous systems.
Definition of Homogeneous Systems:
- A system of linear equations is called homogeneous if it can be written in the form Ax = 0, where 'A' is a matrix and 'x' is the vector of variables. The vector 'b' is the zero vector.
- Mathematically, a homogeneous system is represented as Ax = 0.
Definition of Non-Homogeneous Systems:
- A system of linear equations is called non-homogeneous if it can be written in the form Ax = b, where 'b' is a non-zero vector.
- Mathematically, a non-homogeneous system is represented as Ax = b, where b ≠ 0.
Zero Vector Solution:
- The zero vector, x = 0, is always a solution to the homogeneous equation Ax = 0.
- For the system Ax = b, if the zero vector x = 0 is a solution, then substituting x = 0 into the equation gives A(0) = b, which simplifies to 0 = b. Therefore, 'b' must be the zero vector for the zero vector x = 0 to be a solution.
The statement claims that Ax = b is not homogeneous if the zero vector is a solution. However, for the zero vector to be a solution, 'b' must be zero, making the system homogeneous.
Therefore, I disagree with the statement. The equation Ax = b is homogeneous if the zero vector is a solution to that system, because this implies 'b' must be the zero vector, confirming that the system is indeed homogeneous.
