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Let A be a square n x n matrix. The following statements are equivalent. That is, for a given A, the statements are either all true or all false.

1. A is non-singular.

2. A is row equivalent to In.

3. Ax=0 has only the trivial solution.

4. Ax=b has a unique solution, for each vector b in Rn .

5. Ax=b has at least one solution, for each vector b in Rn .

6. det(A) ≠ 0

7. The column vectors of A form a linearly independent set in Rn.

8. A(transpose) is non-singular.

9. The column vectors of A span Rn.

10. The column vectors of A form a basis for Rn.

I am thinking in the range of topics like basis, dimension, rank, column space, row space, null space, etc.

any help is highly appreciated.