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The following issue comes up whenever I teach linear algebra: I want to have a quick way to say that a vector $(x,y,z)$ is in $\mathbb{R}^3$. I am tempted to say that it has "length $3$". But then some student interprets this as saying that $\sqrt{x^2+y^2+z^2} = 3$. Is there some word other than "length" I could introduce and use to consistently refer to the number of coordinates in a vector? (To point out that I am not nuts here, the commands to get this number in Mathematica and MATLAB are Length[] and length().)

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    $\begingroup$ Would "dimension" do? $\endgroup$ – Jasper Dec 9 '18 at 16:22
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    $\begingroup$ This notion is extrinsic; there is no way to look at an object and tell 'the' vector space it belongs to (and one has stuff like $\mathbb{R}^{2n} \simeq \mathbb{C}^n$). But seconded, maybe 'of real dimension 3' or 'real 3-tuple' might do. $\endgroup$ – Vandermonde Dec 9 '18 at 16:29
  • $\begingroup$ I would also use dimension, but if the vector is known to sit inside some proper subspace I might also use "ambient dimension" to indicate how many components it has. $\endgroup$ – Adam Dec 9 '18 at 16:44
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    $\begingroup$ @user683: $n$-vector has the unfortunate collision with certain physics terminology. $\endgroup$ – Willie Wong Dec 10 '18 at 15:02
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    $\begingroup$ More to the point of the question itself: which kind of linear algebra course are you teaching? Is this an abstract linear algebra course, or an engineering one? What is your main perspective in terms of identifying an $n$-dimensional vector space over the field $\mathbb{F}$ with the space of $n$-tuples $\mathbb{F}^n$? I ask because the programming conventions are implicitly fixing a basis and always identifying vectors with tuples, hence the length(). $\endgroup$ – Willie Wong Dec 10 '18 at 15:05
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If all your spaces are $\mathbb{R}^n$ then you can say n-dimensional (three-dimensional for $\mathbb{R}^3$). However the space of matrices $\displaystyle \left(\begin{array}{cc}a&b\\c&d\end{array}\right)$ ($a,b,c,d\in\mathbb{C}$) is four-dimensional over $\mathbb{C}$, eight-dimensional over $\mathbb{R}$ and infinite-dimensional over $\mathbb{Q}$. So be clear what the base field is. Alternatively you could just write $(a,b,c)\in\left(\mathbb{R}^3,+,\cdot\right)$ or more briefly $(a,b,c)\in\mathbb{R}^3$.

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Along with "dimension", you could also use "component". A vector in three dimensions has three components.

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I say "This is a 3D vector" or "This is a 7D vector".

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  • $\begingroup$ I feel compelled to comment that this is used in a social context in which we are working on some problem involving some vectors in $\mathbb{R}^n$, where these vectors are being thought of as lists of numbers. I certainly wouldn't use this terminology if I was talking about a vector living in an arbitrary vector space. $\endgroup$ – Steven Gubkin Dec 11 '18 at 1:41

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