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Problems about Subspaces

  1. Jun 5, 2012 #1
    1. The problem statement, all variables and given/known data

    -Problem number 1

    Given the set {u ,v} , where u=(1,2,1) and v=(0,-1,3) in R^3 find an equation for the space generated by this set.

    -Problem number 2

    The subspace S is defined as S= {(x,y,z) : x + 2y - z =0}
    find a set B={u,v} in R^3 such that each vector in S is a linear combination of vectors in B.

    2. Relevant equations



    3. The attempt at a solution

    I have no idea how to solve problem number 2.

    I don't know how to find an equation for the space in problem one

    I started the problem like this (a,b,c) =(1,2,1)x + (0,-1,3)y

    and then I found x and y in terms of a and b. but I don't have an idea what is meant to find an equation for the space.

    I would appreciate some help, thanks a lot.
     
  2. jcsd
  3. Jun 5, 2012 #2
    1. How about su+tv for real s and t?
    2. Find two vectors in the plane. To do this, try z=0, et cetera...
     
  4. Jun 5, 2012 #3
    the answer for the first problem is 7x -3y -z = 0 but I don't know how to solve for that

    what do you mean s and t?
     
  5. Jun 5, 2012 #4
    1. s and t would give the parametrized version of the plane. To get the implicit version they give, try the cross product, which gives the normal.
     
  6. Jun 5, 2012 #5

    vela

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    If you have three variables and only one equation, you can solve for one variable in terms of the others. For example, if you had x-2y-3z=0, you could solve for x and get x=2y+3z. Now let y=s and z=t, where s and t are your free parameters, so you have
    \begin{align*}
    x &= 2s + 3t \\
    y &= s \\
    z &= t
    \end{align*} Can you see how to write those three equations as one vector equation?
     
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