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Invertibility of Matrices

Can you please explain to me why the columns of the nxn matrix A span R^n when A is invertible? I feel that if matrix A has columns that span R^n, then the inverse of A should likewise share that same characteristic, the spanning. But I'm not sure if that is a sufficient relationship. Can you give an example of matrix A spanning when it is invertible, if possible?
How does Theorem 4 relate to this?

Theorem 4 states:

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.

Solution Summary

Invertibility of Matrices is investigated. The solution is detailed and well presented.