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().)
$\begingroup$
$\endgroup$
8
3 Answers
$\begingroup$
$\endgroup$
Along with "dimension", you could also use "component". A vector in three dimensions has three components.
$\begingroup$
$\endgroup$
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 $(a,b,c)\in\mathbb{R}^3$.
$\begingroup$
$\endgroup$
1
I say "This is a 3D vector" or "This is a 7D vector".
answered Dec 9, 2018 at 16:34
-
$\begingroup$ 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. $\endgroup$ Dec 11, 2018 at 1:41
length(). $\endgroup$