Generally speaking, Ax=b considers more about the rows of A while Ax=0 more about the columns.
Existence:
1. Ax=b is consistent: (there are no all-zero rows) rank of matrix = rank of augmented matrix
2. Ax=0 is consistent: always (x=0)
Uniqueness:
1. Ax=b
2. Ax=0 has a infinity of solutions: it has at least one free variable
Subspace:
If A is m by n, [a1, ... ,an] are the columns of A
1. Col A
All the linear combinations of columns of A (i.e. b where Ax=b)
In fact it is Span={a1, ... ,an}
It is a subspace of Rm
2. Null A
The solutions of Ax=0
It is a subspace of Rn
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