Let A be an mxn matrix. Then the following statements are logically equivalent. That is, for a particular A, either they are true statements or they are all false.
a. For each b in R^m, the equation Ax=b has a solution.
b. Each b in R^m is a linear combination of the columns of A.
c. The columns of A span R^m.
d. A has a pivot position in every row.