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Homework Help: Prove that V is a subspace of R4

  1. Apr 4, 2010 #1
    1. The problem statement, all variables and given/known data
    Prove that V is a subspace of R4

    Actual problem is attached


    2. Relevant equations

    - S contains a zero element
    - for any x in V and y in V, x + y is in V
    - for any x in V and scalar k, kx is in V

    3. The attempt at a solution

    Its obvious that V is a subset of R4. And I know I must prove that is it contains zero vector which is obvious. But how to I prove its closure, I can see that it is by looking but how exactly do I denote it?
     

    Attached Files:

  2. jcsd
  3. Apr 4, 2010 #2

    Office_Shredder

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    If x and y are in V, that means that Ax=0 and Ay=0. Can you prove that A(x+y)=0? And can you prove that A(tx)=0 for every t a scalar? To prove that V is a subspace you need to prove that for every x,y in V, x+y and tx are also in V, which is precisely what the above shows
     
  4. Apr 4, 2010 #3
    So do these using the 0 vector?
     
  5. Apr 4, 2010 #4

    Office_Shredder

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    Where does the 0 vector come in here? You're supposed to be looking at x and y any vectors in V. So the only property you can use about them is that Ax and Ay are both 0
     
  6. Apr 4, 2010 #5

    Mark44

    Staff: Mentor

    Your set V is all the vectors x in R4 such that Ax = 0. Clearly A0 = 0, but there are other vectors in V, and you need to check that addition is closed for these vectors and scalar multiplication is closed also.
     
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