The discussion revolves around the consistency of linear systems represented by the equation Ax = b, where A is an m*n matrix and b1, b2 are m*1 vectors. The original poster questions whether the system Ax = b1 + b2 is necessarily consistent given that both Ax = b1 and Ax = b2 are consistent.
Some participants have offered insights into the reasoning behind the problem, discussing the properties of linear combinations of solutions. However, there is still uncertainty regarding the completeness of the proof and whether the reasoning fully addresses the original question.
Participants express a desire for clarity and certainty in the proof, indicating a concern about the validity of their reasoning and the existence of multiple solutions to the equations involved.
iamzzz said:Homework Statement
Suppose A is m*n matrix b1 and b2 are m*1 vector and the systems Ax=b1 and Ax=b2 are consistent. Is the system Ax=b1+b2 necessarily consistent?Homework Equations
The Attempt at a Solution
I think Ax = b1 + b2 should be consistent but i don't know how to prove..
Dick said:No thoughts about how to prove at all? A(x1+x2)=Ax1+Ax2. Think about it some more.
iamzzz said:let x1 be the solution of Ax1=b1 and x2 be the solution of Ax2=b2
So Ax1+Ax2=b1+b2=A(x1+x2)
so (x1+x2)could be x
prove ?
Is this correct ?
I mean does that prove the problem ?Dick said:I would say let x1 be ANY solution to Ax=b. There may be more than one. But why are you asking "Is this correct?". What part of it are you worried about?
iamzzz said:I mean does that prove the problem ?
Anyway thanks for the help