Question

True or False?
If Ax=0 has infinitely many solutions then Ax=b will have no solution or infinitely many solutions but not a unique solution.

Answer 1

Let say \(x_1\) is a solution to Ax =b
and by the first statement, there are \(x_2, x_3, \), etc
that are solutions to Ax=0
notice that
\[ A(x_1 + x_2) = Ax_1 +Ax_2 = b +0 = b \]
thus x = \( x_1+x_2\) is also a solution to Ax=b
and because there are an infinite number of solutions to Ax=0
i.e. lots of choices for \(x_2\) , there are an infinite number of solutions for Ax=b
if, on the other hand it is possible there is no solution for Ax=b

so the possibilities are either (1) no solution , or (2) infinite number of solutions