# Show that if Ax=b has more than 1 solution, it has infinitly many solutions

• Jan 25th 2011, 04:49 PM
Jskid
Show that if Ax=b has more than 1 solution, it has infinitly many solutions
Show that if Ax=b has more than 1 solution, it has infinitly many solutions.

Hint: If \$\displaystyle u_1\$ and \$\displaystyle u_2\$ are two solutions consider \$\displaystyle w=ru_1+su_2\$ where \$\displaystyle r+s=1\$

Here's my start
let \$\displaystyle u_1\$ and \$\displaystyle u_2\$ be distinct solutions to \$\displaystyle Ax=b\$ then \$\displaystyle Au_1=b\$ and \$\displaystyle Au_2=b\$

Why must r+s=1?
• Jan 25th 2011, 06:36 PM
mr fantastic
Quote:

Originally Posted by Jskid
Hint: If \$\displaystyle u_1\$ and \$\displaystyle u_2\$ are two solutions consider \$\displaystyle w=ru_1+su_2\$ where \$\displaystyle r+s=1\$

Here's my start
let \$\displaystyle u_1\$ and \$\displaystyle u_2\$ be distinct solutions to \$\displaystyle Ax=b\$ then [maht]Au_1=b[/tex] and \$\displaystyle Au_2=b\$

Why must r+s=1?

\$\displaystyle A(r u_1 + su_2) = b (r + s)\$ ....