Prove b is in the span of the columns of A iff the system [A | b] has a solution.
Prove the columns of A are linearly dependent iff the system [A | 0] has a nontrivial solution.
Prove if A is invertible then Ax=b has a unique solution.
Prove if A is invertible then Ax=0 has only the trivial solution.
Prove if A is invertible and has n columns then col(A) is R^n.
Prove a square matrix is invertible iff its columns are linearly independent (may assume any of the above).