Question

Choose either true or false for each statement

a. There is a vector [b1 b2] so that the set of solutions to

1	0	1
0	1	0

[ x1, x2 , x3,] =[b1b2] is the z-axis.  

b. The homogeneous system Ax=0 has the trivial solution if and only if the system has at least one free variable.

c. If x is a nontrivial solution of Ax=0, then every entry of x is nonzero.

d. The equation Ax=b is homogeneous if the zero vector is a solution.

e. A homogeneous system is always consistent

f. The solution set of a consistent inhomogeneous system Ax=b is obtained by translating the solution set of Ax=0.

Answer 1

a. FALSE. The solution to the equation A (x₁, x₂ , x₃)^T =(b₁,b₂)^T is (x₁, x₂ , x₃)^T =(b₁-x₃, b₂, x₃)^T . Whatever be the value of b₁,b₂, no solution of the equation can be the vector (0,0,x₃)^T i.e. the z-axis or, the x₃ -axis.

b. FALSE. The homogeneous system Ax = 0 always has the trivial solution.

c. FALSE. If x is not equal to the zero vector, and Ax = 0, then x is a non-trivial solution.

d. TRUE. If 0 is a solution, then b = A0 = 0 for any matrix A.

e. TRUE. It always has the trivial solution.

f. TRUE.