Mark each statement True or False. Justify each answer.
a. A homogeneous system of equations can be inconsistent.
b. If x is a nontrivial solution of Ax = 0, then every entry in x is non zero.
c. The effect of adding p to a vector is to move the vector in a direction parallel to p.
d. The equation Ax = b is homogeneous if the zero vector is a solution.
e. If Ax = b is consistent, then the solution set of Ax = b is obtained by translating the solution set of Ax = 0.
a. False, the trivial solution is always a solution: consistent.
b. False, you could create a 3x3 matrix A with top two rows containing 1’s and bottom row can be anything, a non-trivial solution would be (1 -1 0) (transposed).
c. True, it is like sliding the vector along p.
d. True, since Ax = A0 = 0, b must be the zero vector and thus the system is homogeneous.
e. I don’t have an explanation for this onze, but I think it is true.