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If-a-vector-v-exists-in-n-dimensions-v-R-n-Can-there-exist-a-complex-dimension-s-




Question Number 6932 by FilupSmith last updated on 03/Aug/16
If a vector v exists in n dimensions:  v∈R^n   Can there exist a complex dimension(s)?
$$\mathrm{If}\:\mathrm{a}\:\mathrm{vector}\:\boldsymbol{{v}}\:\mathrm{exists}\:\mathrm{in}\:{n}\:\mathrm{dimensions}: \\ $$$$\boldsymbol{{v}}\in\mathbb{R}^{{n}} \\ $$$$\mathrm{Can}\:\mathrm{there}\:\mathrm{exist}\:\mathrm{a}\:\mathrm{complex}\:\mathrm{dimension}\left(\mathrm{s}\right)? \\ $$
Commented by nburiburu last updated on 04/Aug/16
you mean if v∈R^n  ⇒v∈C^n ?  really, it could be exist afunction that  transform v from one space to the other  but they are different dimension spaces  so you can never say they are the same  (as equal vector), but you can say they are  the same (by applying a linear transformation)
$${you}\:{mean}\:{if}\:{v}\in\mathbb{R}^{{n}} \:\Rightarrow{v}\in\mathbb{C}^{{n}} ? \\ $$$${really},\:{it}\:{could}\:{be}\:{exist}\:{afunction}\:{that} \\ $$$${transform}\:{v}\:{from}\:{one}\:{space}\:{to}\:{the}\:{other} \\ $$$${but}\:{they}\:{are}\:{different}\:{dimension}\:{spaces} \\ $$$${so}\:{you}\:{can}\:{never}\:{say}\:{they}\:{are}\:{the}\:{same} \\ $$$$\left({as}\:{equal}\:{vector}\right),\:{but}\:{you}\:{can}\:{say}\:{they}\:{are} \\ $$$${the}\:{same}\:\left({by}\:{applying}\:{a}\:{linear}\:{transformation}\right) \\ $$

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