# Word for the dimension of the vector space in which a vector lives?

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().)

• Would "dimension" do? – Jasper Dec 9 '18 at 16:22
• 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. – Vandermonde Dec 9 '18 at 16:29
• 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. – Adam Dec 9 '18 at 16:44
• @user683: $n$-vector has the unfortunate collision with certain physics terminology. – Willie Wong Dec 10 '18 at 15:02
• 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(). – Willie Wong Dec 10 '18 at 15:05

## 3 Answers

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 $$(a,b,c)\in\mathbb{R}^3$$.

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

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

• 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. – Steven Gubkin Dec 11 '18 at 1:41