39. Suppose A is an m n matrix with the property that for all b in Rm the equation Ax D b has at most one solution. Use the definition of linear independence to explain why the columns of A must be linearly independent.

Answer: trivial solution is the only solution to Ax = 0, therefore columns must be linearly independent.

Step-by-step explanation: Recall that the columns of A are linearly independent if and only Ax = 0 had only the trivial solution. Since Ax = b has at most one solution, therefore the only solution to Ax = 0 is the trivial solution and the column vectors must be linearly independent.