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Verify Subset is a Subspace

  1. Jul 5, 2009 #1
    Okay then. I just read the section of Axler on subspaces. It says that if U is a subset of V, then to check that U is a subspace of V we need only check that U satisfies the following:

    additive identity

    [tex]0\in U[/tex]

    closed under addition

    [tex]u,v\in U\text{ implies }u+v\,\in\,U[/tex]

    closed under scalar multiplication

    [tex]a\in\mathbf{F}\text{ and }u\in U\text{ implies }au\in U[/tex]

    Now I am supposed to use these axioms to verify that for [itex]b\in\mathbf{F}[/itex], then

    [tex]{(x_1,x_2,x_3,x_4)\in\mathbf{F}^4:x_3=5x_4+b}[/tex]

    is a subspace of F4 iff b=0.

    I am not exactly sure how to actually apply those 3 axioms to this problem?

    How does one test that [itex]u,v\in U\text{ implies }u+v\,\in\,U[/itex]?

    Should I start with something like:

    (x1,x2,x3,x4)+(y1,y2,y3,y4)=(x1+y1,x2+y2,5x4+b+5y4+b,x4+y4) ?
     
    Last edited: Jul 5, 2009
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  3. Jul 6, 2009 #2

    Office_Shredder

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    [tex](x_1,x_2,x_3,x_4) + (y_1,y_2,y_3,y_4) = (x_1 + y_1, x_2 + y_2, x_3 + y_3, x_4 + y_4)[/tex]

    by definition. So there are two parts to this problem

    1) Show if b=0 then this is a subspace
    2) Show if this is a subspace then b=0

    To work towards 1), if

    [tex]x_3 = 5x_4[/tex] and [tex]y_3 = 5y_4[/tex]

    show the RHS satisfies the necessary relationship. Rinse and repeat for the other stuff

    To do part 2, you need to find a contradiction. So look at the three things required for the subset to be a subspace, and see if all of them hold
     
  4. Jul 6, 2009 #3
    Okay. So just dealing with the x3+y3 part of the addition

    x3+y3=5x4+5y4+2b.. I am sorry, but I still do not see why b must equal 0 :confused: what part of the definition is being violated if it is not?
     
  5. Jul 6, 2009 #4
    Check the additive identity first. What is the additive identity? Now what happens if b is not 0?
     
  6. Jul 6, 2009 #5
    Okay, I see now. I forgot about the other properties for a moment.

    Can someone help me out wit the formalism of this? If this were a HW problem, my 'proof' would need to follow a certain format. Should I name the proposed subspace and then continue. Like this.

    Proposition:
    [tex]U={(x_1,x_2,x_3,x_4)\in\mathbf{F}^4:x_3=5x_4+b)\text{ is a subspace of }\mathbf{F}^4\text{ iff }b=0
    [/tex]

    Proof:




    Now I am not sure how to formally state that since a subspace must include the additive identity then b must equal 0. How do you math savvy types do this?
     
  7. Jul 6, 2009 #6

    Dick

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    You don't have to be that math savvy. You want to solve the equations x3=0, x4=0 and x3=5*x4+b. For what values of b is that possible?
     
  8. Jul 6, 2009 #7
    Hey Dick. I realize that I don't have to be math savvy. I want to know what the typical manner in which one presents the verification of a problem such as this.

    That way in the future I won't hit snags in proofs with the simple mechanics of presenting it. I will just have to worry about the logic.
     
  9. Jul 6, 2009 #8

    Dick

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    Well, as I said, present it as a problem of whether the linear equation has the solution (x1,x2,x3,x4)=(0,0,0,0), which is what you need to have the additive identity in the subset.
     
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