Question

Suppose ax = 0 has only the trivial solution. Explain why the equation ax = b has the unique solution.

Answer 1

Starting from the assumption that Ax=b has a unique solution, suppose that Ax=0 had a nontrivial solution, i.e. Ax¯=0 for some x¯≠0. Then, assuming x∗ is the solution of Ax=b:

Starting from the assumption that Ax=b has a unique solution, suppose that Ax=0 had a nontrivial solution, i.e. Ax¯=0 for some x¯≠0. Then, assuming x∗ is the solution of Ax=b:A(x∗+x¯)=Ax∗+A(x¯)=b+0=b

Starting from the assumption that Ax=b has a unique solution, suppose that Ax=0 had a nontrivial solution, i.e. Ax¯=0 for some x¯≠0. Then, assuming x∗ is the solution of Ax=b:A(x∗+x¯)=Ax∗+A(x¯)=b+0=bwhich would show that Ax=b has at least two solutions, a contradiction.

OR

If Ax=b has two solutions x1,x2 or

If Ax=b has two solutions x1,x2 orAx1=bAx2=b

If Ax=b has two solutions x1,x2 orAx1=bAx2=bthen by linearity

If Ax=b has two solutions x1,x2 orAx1=bAx2=bthen by linearityA(x1−x2)=b−b=0

If Ax=b has two solutions x1,x2 orAx1=bAx2=bthen by linearityA(x1−x2)=b−b=0but the solution for A(x1−x2)=0 is x1−x2=0⇒x1=x2