Ax=b1 and Ax=b2 are consistent. Is the system Ax=b1+b2 necessarily consistent?

  1. 1. The problem statement, all variables and given/known data
    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?


    2. Relevant equations



    3. The attempt at a solution

    I think Ax = b1 + b2 should be consistent but i dont know how to prove..
     
  2. jcsd
  3. Dick

    Dick 25,853
    Science Advisor
    Homework Helper

    No thoughts about how to prove at all? A(x1+x2)=Ax1+Ax2. Think about it some more.
     
  4. 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 ?
     
  5. Dick

    Dick 25,853
    Science Advisor
    Homework Helper

    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?
     
  6. I mean does that prove the problem ?

    Anyway thanks for the help
     
  7. Dick

    Dick 25,853
    Science Advisor
    Homework Helper

    I'm just saying I would feel better if you KNEW it solved the problem instead of having to ask. Yes, it's fine.
     
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