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Is the following a basis?

  1. Oct 28, 2012 #1
    Suppose that S = {v1, v2, v3} is a basis for a
    vector space V.
    a. Determine whether the set T = {v1, v1 +
    v2, v1 + v2 + v3} is a basis for V.
    b. Determine whether the set
    W = {−v2 + v3, 3v1 + 2v2 + v3, v1 −
    v2 + 2v3} is a basis for V.


    I must check if they're linearly independent.

    For a:
    c1v1+c2v1+c2v2+c3v1+c3v2+c3v3=0 c's are constants
    v1(c1+c2)+v2(c2+c3)+v3(c3)
    Forming the matrix gives
    1 1 0
    0 1 1
    0 0 1
    rref of this matrix is the identity matrix, thus it's linearly independent.


    For b:
    the same thing was done except the rref of the matrix was not the identity matrix, thus it's not a basis.


    My question is is there an easier way to do this problem? It seems i made it harde/longer.
     
  2. jcsd
  3. Oct 28, 2012 #2

    LCKurtz

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    It would be clearer if you start$$c_1v_1+c_2(v_1+v_2)+c_3(v_1+v_2+v_3)=0$$
    and maybe you wouldn't have made that mistake, should be
    $$(c_1+c_2+c_3)v_1+(c_2+c_3)v_2 +c_3v_3=0$$
    Yes, you can do it in your head. Looking at my last equation you see ##c_3=0##. Then since ##(c_2+c_3)=0## you know ##c_2=0## so...

    Sometimes the equations are so simple it isn't worth the time to row reduce.
     
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