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Vector Space

  1. Oct 28, 2014 #1
    • OP warned about not using the homework template
    Let B be a non-zero mx1 matrix, and let A be an mxn matrix. Show that the set of solutions to the system AX=B is not a vector space.

    I am thinking that I need to show that the solution is not consistent. In order to do so would I need to show that B is not in the column space of A?
     
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  3. Oct 28, 2014 #2

    Dick

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    It's a lot simpler than that. A vector space is supposed to contain a zero vector.
     
  4. Oct 28, 2014 #3
    So unless B is a zero vector than in any case AX=B can not be a vector space? It must be AX=0, or the null space?
     
  5. Oct 28, 2014 #4

    Dick

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    That's kind of confusing, but if X=0 doesn't solve your system then the set of solutions isn't a vector space.
     
  6. Oct 28, 2014 #5
    Sorry, I am still trying to wrap my head around vector spaces. Could you elaborate why X=0 must solve the system to be a vector space.
     
  7. Oct 28, 2014 #6

    Dick

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    When does a set of vectors constitute a vector space? You'll need to look up the definition if you can't recall. The answer is there.
     
  8. Oct 28, 2014 #7

    HallsofIvy

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    If x and y both satisfy Ax= B, Ay= B, then A(x+ y)= Ax+ Ay= B+ B= 2B. Unless B= 0, x+ y does NOT satisfy the equation A(x+ y)= B so is NOT in this set. The set is not closed under addition, so is not a vector space.
     
  9. Oct 28, 2014 #8
    1) Vector addition of vectors u,v
    2) Scalar multiplication of a real number a, and u
    3) A vector space has to have a zero vector
     
  10. Oct 28, 2014 #9
    Since AX=B does not satisfy 3 because if X=0 then B must be a zero vector then it is not a vector space. Am I correct?
     
  11. Oct 28, 2014 #10

    Dick

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    Yes, you are correct. Some other properties don't work either as Halls pointed out. But that's the easiest one to check.
     
  12. Oct 28, 2014 #11
    Much appreciated for the help from the both of you.
     
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