Explanation: A linear system Cx = d is known as a homogeneous system if d! = 0. The homogeneous linear system Ax = 0 is called its corresponding homogeneous linear system.

Explanation: In an m-dimensional space, every linear equation produces a hyperplane for n variables. The solution set is the intersection of these hyperplanes and is planar which may have any dimension smaller than m.

Explanation: Let i be the number of equations and j be the number of unknowns in the given system. Since i> j, the system has at least one solution x1 = 0, x2 = −6, and x3 = 4 and so it is consistent. Thus, it results in either a unique solution or infinitely many solutions.

Explanation: Since m<n, the system is either inconsistent or has infinitely many solutions. Since y1 = 6, y2 = −4, y3 = 0 is a solution of the system, the system is not inconsistent. Thus the only possibility is infinitely many solutions.

Explanation: Since the rank of this homogeneous system(which is always consistent) and the number of unknowns are equal, the only possible solution is zero and it is a unique solution.

Explanation: The possibilities for the solution set for any homogeneous system is either a unique solution or infinitely many solutions. Since the homogeneous system has the zero solution and y1 = 5, y2 = −3, y3 = 2 is another solution, it has at least two distinct solution. Thus the only possibility is infinitely many solutions.

Explanation: Since m1 = 4, x2 = −7, x3 = 0 is a solution of the system, the system is not inconsistent. Thus the only possibility is infinitely many solutions.

Explanation: Here the number of equations and the number of unknowns are equal and the system is homogeneous, so it may have the zero solution or infinitely many solutions.

Explanation: Since the system is homogeneous and there are more equations than the number of unknowns, so the possibilities are either a unique solution or infinitely many solutions. However, if the rank r of the system is 5, then it can be a unique solution as well as if r<5, then there are infinitely many solutions.

Explanation: A homogeneous system is consistent. The rank is r = 3 and the number of variables is n = 4. Hence there is n – r = 1 free variable. Thus there are infinitely many solutions.