Math, Linear Algebra Question?

Question

Say whether each of the following statements is true or false, and give a brief reason in each case.
(a) Every system of 3 linear equations in 5 unknowns has many solutions.
(b) If a system Cx = b has a solution x, then the columns of C are dependent.
(c) If two matrices D and E reduce to the same echelon form then D = E.

Answer 1

a) False. Consider the system a+b+c+d+e=0, b+c+d+e=1, and a=0. This system has no solutions.
b) False. In fact, if the system has a unique solution, the columns of C cannot be dependent, since that would imply C is not invertible.
c) False. Consider the following matrices:
[1, 0]
[0, 1]
and
[1, 0]
[1, 1]
These have the same rref, but are not equal.

Edit: decided the previous formatting was too ambiguous

Answer 2

(a)False. If you have 3 linear equations and 5 unknowns in a system, you first of all can't solve for all 5 solutions, except in certain cases. Also, there is a great possibility that a system of 3 equations will have no solution at all, becuase it isn't necessary that 3 lines intersect at one single point at all.

(b)I'm not sure on this one.

(c)False, since the same echelon doesn't necessarily mean that they have the same stored values. For example, if one had 4, and the other had 2, they could both be reduced to 2 even though they are different values.

Answer 3

(a) is true. If you have more unknowns than equations, you have many solutions. If you have the same number of equations as unknows, you have one solution.

Example: 2 unknowns, 1 equation. x + y = 7. solutions: (1, 6), (2, 5), etc.