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

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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?
 

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  • #2
Simon Bridge
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The question is: are x and ix linearly dependent or independent?
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.

when you solve ax+ibx=0 (where x≠0), you get a=-ib which shows that the only solution can be a,b=0.
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?
 
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pwsnafu
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The question is: are x and ix linearly dependent or independent?
In what vector space, with what field?

My first guess is that they should be linearly dependent since i is a constant.
Linear dependence has nothing to do with whether something 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.
Why? ##b=2## and ##a=-2i## is also a solution.

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

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