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Here's an equation from a text book for computing a unit vector: $$\hat v = \frac{\overline v}{ \sqrt{ \sum ^n _{i=1} (\overline v_i)^2 } }$$

Now I may be wrong here, but using $n$ doesn't really cut it here for me. $n$ can't be any old amount, it has to be the number of dimensions in $ \overline v$, right? So is there a symbol for the number of dimensions in a vector?

Starkers
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    I suspect that it is implicit that $n$ is the dimension of the space that $\bar{v}$ 'lives' in. Perhaps it states $\bar{v} \in \mathbb{R}^n$ somewhere? – copper.hat May 29 '14 at 15:57
  • Sometimes we say that if $v \in V$, the $n = \dim V$. So if you really want to take the sum over all the dimensions, you would write $\sum_{i=1}^{\dim V} (\bar v_i)^2$. – Tyler Holden May 29 '14 at 16:00
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    Note that the notation not only relies on fixing the dimension of the vector space, but also a particular basis! – Hagen von Eitzen May 29 '14 at 16:00
  • @copper.hat It genuinely doesn't, but that does make sense. Could you briefly explain that ∈ please? – Starkers May 29 '14 at 16:24
  • It means 'an element of'. The symbol $\mathbb{R}^n$ refers to the collection of $n$-dimensional vectors whose components are real. – copper.hat May 29 '14 at 16:30

2 Answers2

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Notation typically identifies a space's dimension, not a vector's space's dimension; we write $\dim V$, not $\dim\operatorname{Space}v$ or anything like that. If $n$'s inclusion here bothers you, I recommend rewriting the sum, e.g. as $v^\ast\cdot v$ or $\sum_i\overline{v}_i^2$. (Without explicit constraints on $i$, $\sum_i$ means "sum over all relevant $i$", for a suitable definition of relevance that's obvious in context).

J.G.
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Implicit in the question is that the vectors are in $\mathbb{C}^n$, which is a particular type of vector space. There are many others, e.g. various types of function space, polynomial space, matrix space. None of these have "coordinates" as such in their representations -- this exists only in coordinate spaces such as $\mathbb{R}^n$ or $\mathbb{C}^n$. However to have a coordinate space you must specify how many coordinates there are; hence the notation requested is superfluous.

Note: obviously some authors omit specifying what the ambient space is such as apparently the author of the text in question; however using the letter $n$ for the dimension of the coordinate space is so common as to be considered standard.

vadim123
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