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Are x and ix linearly dependant or independant? (i=√-1)

  1. Oct 11, 2013 #1
    The question is: are x and ix linearly dependent or independent?

    My first guess is that they should be linearly dependent since i is a constant.

    But when you apply the definition of linear independence i.e. when you solve ax+ibx=0 (where x≠0), you get a=-ib which shows that the only solution can be a,b=0.

    Hence, according to definition of linear independence, x and ix should be linearly independent.

    Am I correct?
  2. jcsd
  3. Oct 11, 2013 #2

    Simon Bridge

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    I think you need to be claer about how you are setting up the vector space.
    i.e. In the complex plane, ix is perpendicular to x.

    Only if you insist that a and b are both real.
    The definition applies over a subset of a vector space - which vector space do these two numbers belong to?
  4. Oct 11, 2013 #3


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    In what vector space, with what field?

    Linear dependence has nothing to do with whether something is a constant.

    Why? ##b=2## and ##a=-2i## is also a solution.

    Consider the complex numbers as a one dimensional vector space over itself. What happens?
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