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>Since complex numbers are also a vector space (of dimension 1) This statement seems very likely to confuse someone who doesn't know about or understand how to
by cool_dude85 2y ago
>Since complex numbers are also a vector space (of dimension 1)
This statement seems very likely to confuse someone who doesn't know about or understand how to compute the norm of a complex number.
Complex numbers are a 1-dimensional vector space over the complex numbers themselves, but are probably more readily understood as a two-dimensional vector space over the reals. Any field is a vector space of dimension 1 over itself.
- youoy 2y agoHe does this to justify that we can define a norm over C, because the definitions speaks about vector spaces. For me that section is not necessary, because he justifies the formula for the norm based on the definition of the norm, which in turn is historically justified by the norm over R2, which is by no doubt the first example of norm discovered (Pythagoras theorem).
- phkahler 2y ago>> He does this to justify that we can define a norm over C Yeah, but computing z* really involves taking apart the complex number and fiddling with the real coefficient of the imaginary part. zz* looks like an operation in C, but it's kind of not.
- tczMUFlmoNk 2y agoFor a real number `x`, does computing `-x` require "taking apart the real number and fiddling with the sign bit"? I think of negation as a real operation in R, namely a reflection about the origin, and conjugation a a real operation in C, namely a reflection about the real axis.