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Homework Help: Finding a basis for a particular subspace with Dot Product restrictions

  1. Nov 12, 2014 #1
    Find the basis of the subspace of R4 that consists of all vectors perpendicular to both [1, -2, 0, 3] and [0,2,1,3].

    My teacher applies dot product: Let [w,x,y,z] be the vectors in the subspace. Then,

    w-2x+3z=0 and 2x+y+3z=0

    So, she solves the system and get the following:

    Subspace= { t[-1,-1/2,1,0] + s[-6,-3/2,0,1]|t,s are in R}.

    But, I do the following:

    I isolate w and y: w=2x-3z and y=-2x-3z.

    I replace them : Supspace= { [2x-3z,x,-2x-3z,z]|x,z are in R} = span{[2,1,-2,0],[-3,0,-3,1]}.

    I set up a system of linear equation to see if [-3,0,-3,1] is a linear combination of the vectors in my teacher's answer. However, it is not.

    What am I doing wrong?
  2. jcsd
  3. Nov 12, 2014 #2


    User Avatar
    Homework Helper

    It is a linear combination of the teacher's answer.
    You did not do anything wrong with your solution.
    Let t=-3, s=1.
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