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?