Determine whether the following statements are true or false. Justify your answers.
a. The solution set to a system of three equations in three unknowns cannot be a plane. b. A system of linear equations cannot have only two solutions. c. The solution set to a consistent rank 2 linear system in four unknowns would be a line in four-dimensional space. d. A system of four equations in four unknowns always has a solution. e. A system of four equations in four unknowns can have at most one solution.
The Correct Answer and Explanation is :
Let’s evaluate each statement regarding systems of equations:
a. The solution set to a system of three equations in three unknowns cannot be a plane.
Answer: True.
Justification: In a three-dimensional space, the solution set of three equations can be a point, a line, or a plane. However, for three equations in three unknowns to represent a plane, they must be linearly dependent, which is typically not the case. Therefore, the solution set will generally be a unique point or no solution, but not a plane.
b. A system of linear equations cannot have only two solutions.
Answer: True.
Justification: A system of linear equations can either have no solution, exactly one solution, or infinitely many solutions, depending on the relationships between the equations. If there are two solutions, it implies that there is an entire line of solutions, which contradicts the nature of linear equations. Thus, having exactly two solutions is not possible.
c. The solution set to a consistent rank 2 linear system in four unknowns would be a line in four-dimensional space.
Answer: False.
Justification: A consistent rank 2 linear system in four unknowns corresponds to two independent equations. This system can define a plane in four-dimensional space, not just a line, as it will have infinitely many solutions along a plane defined by two dimensions.
d. A system of four equations in four unknowns always has a solution.
Answer: False.
Justification: A system of equations does not guarantee a solution. The system can be inconsistent, meaning that the equations conflict with each other. An example would be two equations representing parallel planes that do not intersect, resulting in no solution.
e. A system of four equations in four unknowns can have at most one solution.
Answer: True.
Justification: If a system of four equations is consistent and the coefficient matrix is of full rank (rank 4), then it has exactly one solution. However, if the system is consistent and underdetermined (fewer independent equations than unknowns), it can have infinitely many solutions. But, under the assumption of consistency and independence, it can indeed have at most one solution.
In summary, the evaluations yield:
a) True, b) True, c) False, d) False, e) True. The reasoning highlights the linear algebra principles guiding these conclusions, focusing on dimensions, rank, and relationships between equations.