Problem:

Prove

Ax=b has a solution for each b in R^m if and only if the equation A^T x = 0 has only the trivial solution.

Hint: For the forward direction use theorem 1.4.4 to prove that the dimension of the null space pf A^T is zero

Theorem 1.4.4: Let A be an mxn matrix. Then the following statements are logically equivalent. That is, for a particular A, either they are all 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