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I'm taking a linear algebra course this semester, and so far we've only talked about vectors in $\mathbb{R}^2, \mathbb{R}^3$ and higher dimensions.

Does it makes sense to talk about one-dimensional vectors in $\mathbb{R}^1$?

Since we visualize $\mathbb{R}^2$ as a plane, would $\mathbb{R}$ be a simple number line?

If one-dimensional vectors are a thing, what are they used for?

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    $\begingroup$ "one-dimensional vector" is an unlucky formulation beacuse a vector has no dimension, but a number of components. So, what you mean is a vector with one component, this behaves like a real number. Such a vector makes sense and is particular easy to handle. $\endgroup$
    – Peter
    Sep 25, 2019 at 19:20
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    $\begingroup$ By the way, a real number can also be considered as a $1\times 1$-matrix. $\endgroup$
    – Peter
    Sep 25, 2019 at 19:23
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    $\begingroup$ Sometimes it can make sense to view the complex numbers as 1d vector space over the reals and then consider $\mathbb{R}$-linear maps. $\endgroup$
    – hal4math
    Sep 25, 2019 at 20:06
  • $\begingroup$ @Peter I don't have a problem with the formulation "$n$-dimensional vector". This seems to be a more concise phrasing of "an element of an $n$-dimensional vector space", which is an entirely reasonable object to discuss. Moreover, not all vectors have "components"---consider, for example, an element of $L^2(\mathbb{R})$, which is a vector, but which is not generally represented in terms of "components". I don't think that it is entirely unreasonable to state that an element of $L^2(\mathbb{R})$ is an "infinite dimensional vector". $\endgroup$
    – Xander Henderson
    Jul 14 at 14:38

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Yes. Not only are one dimensional vectors a thing, "zero dimensional" vectors are too! An example of a one dimensional vector would just be any real number, as you observed. A zero dimensional vector would be an element of the trivial vector space $\{0\}$.

This might seemingly conflict with uses of "vector" one typically learns in middle/high school, which states that a vector is a quantity with magnitude and direction, and quantities with only magnitude are called scalars. You can still think of elements of $\mathbb R$ this way, in drawing an arrow from $0$ to any $x$ on the real number line and having "right or left" (positive or negative) be your direction. But more generally, this is not actually a distinction we make in math, and the definition of a vector space over a field $K$ is abstract in terms of the defining axioms. The abstract definitions may mean that we call some weird things vectors that are hard to get used to at first, but eventually you'll realise the usefulness of our definitions.

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The answer to your first question is yes; you may, if you wish, call any $a \in \mathbb{R}$ a vector. However, there isn’t much purpose, since they fulfill exactly the same purpose as what you may wish to call a scalar. For instance, the vector $<3>$ and the scalar $ 3 $both produce the same vector if you take their dot product / product with any vector in $\mathbb{R}^n$.

To answer your third question, they aren’t really a ‘thing’ for the exact reason stated above that there is no real use for calling them vectors.

To answer your second question, the space you are considering is indeed the number line.

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    $\begingroup$ I think it's pretty useful that theorems and constructions about real vector spaces also apply to $\mathbb R$. $\endgroup$ Sep 25, 2019 at 19:54

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