Question

A system of linear equations is said to be homogeneous if the constants on the right-hand side are all zero. The system

2x1 − x2 + x3 + x4 = 0

5x1 + 2x2 − x3 − x4 = 0

−x1 + 3x2 + 2x3 + x4 = 0

is an example of a homogeneous system. Homogeneous systems always have at least one solution, namely the tuple consisting of all zeros: (0, 0, . . . , 0). This solution of all zeros is called the trivial solution and any other solution is called nontrivial. It turns out that any underdetermined homogeneous system also has at least one nontrivial solution since there will necessarily be free variables that can be taken to be anything we want. Thus, we can conclude that underdetermined homogeneous systems always have infinitely many solutions! With this knowledge in hand, how many solutions do you expect that the system above has? To confirm your answer, write the augmented matrix of the homogeneous system above, reduce it to reduced row echelon form (RREF), and then solve. Circle your final answer.

Answer 1