1. ## [Matrix] Q about Trivial&NonTrivial System

I just like to check

Trivial solution = linearly independent
= unique = when n x m matrix, n=m=rank

NonTrivial solution = linearly independent
= infinite = row of zeros = n>rank

Also. (True of False)

(a) If the system is homogeneous, every solution is trival.
- False, since if the matrix contains a row of zeros (0 0 ... 0|0), then the system is non-trival.

(b) If the system has a nontrivial solution, it cannot be homogeneous.
- False, since homogeneous system is either unique(trival) or infinite(nontrival). Therefore, nontrivial solution must be a homogenous system.

(c) If there exists a trivial solution, the system is homogeneous.
- True, since homogenous system is either unique(trival) or infinite(nontrival). Therefore, trivial solution must be a homogeneous system.

(d) If the system is consistent, it must be homogenous.
- True, since any system that is not inconsistent is called consistent and consistent either has unique or infinite solutions which is a property of a homogeneous system.

Now assume that the system is homogeneous:

(e) If there exists a nontrivial solution, there is no trivial solution.
- True. A matrix with a nontrivial solution must contain a row of all zeros but no trivial solution must not contain a row of zeros in RREF.(What is no trival soln?)

(f) If there exists a solution, there are infinitely many solutions.
- False. If the system has basic solutions (systems without a row of zeros), the system contains unique solutions.

(g) If there exist nontrivial solutions, the row-echelon form of A has a row of zeros.
- True, since linearly dependent systems requires a row of zeros for parameters, which results infinite solutions.

(h) If the row-echelon form of A has a row of zeros, there exist nontrival solutions.
((((((((((((((((((((I think it is possible. I believe what my Professor said... ??? but y??)))))))))))))))))))

(i) If a row operation is applied to the system, the new system is also homogeneous.
-True. If the starting matrix is a homogenous system then the applied row operation system is always a homogenous system because it does not change property of original matrix.

SORRY, IT IS BIT LONG...