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Show that W is a subspace of R3

  1. May 1, 2012 #1
    ε1. The problem statement, all variables and given/known data

    Let W = R3 be the set W = {(x,y,z)|z=y-2x}. Show that W is a subspace of R3

    2. Relevant equations
    From the equations my teacher gave me I know that if W is to be a subspace it needs to follow:

    1.) U,VεW then (U + V)εW,
    2.)UεW and K(random constant)εW then KUεW


    3. The attempt at a solution
    So I haven't really figured out how to attempt this problem but from what I know is that you need to assign U and V to something. From what Ive done so far I have U = (1,-2,-1) and V = (2,-4,-2). I got those numbers from the z=y-2x equation. Then U+V = (3,-6,-3), which I believe froms the first equation I gave.
    Then I believe I pick a random constant for K, so ill choose 3. So 3U = (3,-6,-3) which proves the second equation.

    Are these the proper steps to solving this problem?
     
  2. jcsd
  3. May 1, 2012 #2
    Also I just realized that I mixed up x and y in when I placed them in U and V, but are the steps still correct?
     
  4. May 1, 2012 #3

    Mark44

    Staff: Mentor

    It must also be true that 0 ε W.
    No, they aren't. You can't pick any particular vectors in W and you can't pick a particular constant. You need to show that
    1) 0 ε W.
    2) For any vectors u and v in W, u + v ε W.
    3) For any constant k and any vector u in W, ku ε W.

    If you are given an arbitrary vector u = <u1, u2, u3> in R3, how can you tell whether that vector is in W?
     
  5. May 1, 2012 #4
    So to show that 0 εW, do i plug (0,0,0) into z=y-2x?

    And if I was given any arbitrary vector I should just be able to plug it into z=y-2x correct?
    I am not given any vectors though, just what is shown. I cannot quite wrap my head around this.
     
  6. May 1, 2012 #5

    Mark44

    Staff: Mentor

    Sort of. A better way to show this is to show that (0, 0, 0) satisfies the equation z = y - 2x.
    Any vector u in W must satisfy this equation.

    Here's how to approach this problem:

    Let u = <u1, u2, u3> be an arbitrary vector in W. From the definition of set W, it must be true that u3 = u2 - 2u1.

    Now take another arbitrary vector v in W. Show that u + v ##\in## W.

    For the third part, show that for any arbitrary real number k, and any vector u ##\in## W, then ku ##\in## W.


     
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