Solving a Problem in My Assignment: X1, X2, and X3

[Soma]

Homework Statement

Let X1 = [(1,2,3)] and Let X2 = [(4,5,6)] be two solutions of the linear system AX = B. Find all solutions X3 of this system, such that X3 ≠ X1 and X3 ≠ X2.

Relevant Equations

None

This is just a small part of a question I have in my assignment and I'm not sure how to solve it, nothing in my eBook or our presentation slides hints at a similar problem, what I tried was I noticed that X1 and X2 have the difference of (3,3,3) and I assume either X3 = (3,3,3) or X3 = (7,8,9) is that right or am I getting it wrong?

 

[fresh_42]

There should be infinitely many possibilities. We know nothing about $A$ or $B$. The only fact we have is what you have already observed: $Ax_1=b=Ax_2 \Longrightarrow A(x_2-x_1)=A(3,3,3)=0$. This means the kernel of $A$ is not trivial and at least one dimensional. Thus the kernel may have all dimensions $1,2,3$ depending on what $A$ and $b$ are.

We only know for sure that all vectors $x_1+\lambda (x_2-x_1)$ are solutions, but we cannot know whether these are all solutions.

 

[Soma]

Thank you so much for the help! I wasn't sure what to do at first but that makes a lot of sense