Suppose AX = 0 has only the trivial soln. Prove that, if....

[buckaroobill]

this proof was confusing me and I think I'm probably going to get one like it on my next in-class quiz, so any help would be appreciated greatly:

Suppose the matrix AX=0 only has the trivial solution. Prove that if AX=B has one solution, then AX=B has just one solution.

 

[galactus]

I suppose you could assume A has an inverse, then \(\displaystyle \L\\x=A^{-1}b\)

Therefore, the only solution is unique.

Now, since Ax=b has a unique solution, then Ax=0 has a unique solution. Since 0

is a solution to Ax=0, the only solution to Ax=0 is 0.

 

[pka]

Suppose that each of C & D is a nontrivial solution then:
\(\displaystyle \L \begin{array}{rcl}
AC = B\quad & \& & \quad AD = B \\
AC - AD & = & 0 \\
A\left( {C - D} \right) & = & 0 \\
& \Rightarrow & C - D = 0 \\
& \Rightarrow & C = D. \\
\end{array}\)

That prove that any solution is unique.