ok this was confusing me so any help is appreicated.
suppose A is a n by n matrix. show that the rank of A is n only if the AX = 0 has just the trivial solution.
Meaning A has n linearly independent coloums vectors.
Thus, they form a basis. Thus, a unique representation.
Since expressing the zero coloum is possible trivially it is the only solution for it must be unique.