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Am I correct that a matrix has two dimensions, and a vector has one dimension?

What is the number of dimensions of a scalar? zero?

Thanks.

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    $\begingroup$ Dimension would be the wrong word, since that refers to the size of the spaces they live in, not the way we arrange their information. The word you want would be rank. $\endgroup$ Apr 6, 2020 at 22:13
  • $\begingroup$ Related: 1, 2 $\endgroup$
    – user1210203
    Sep 24, 2023 at 1:48

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The question is vague, and I am afraid that it can have no formal sense, for instance, because the definitions of scalar and dimension were not provided. In my answer below I addressed these issues, but it is still informal.

First we need to define a scalar. For instance, if we have a vector space $V$ over a field $F$ then we can say that scalars are elements of $F$. This definition is confirmed, for instance, by [M], according to which, in general case scalar is an element of a field. Next, we have to define a dimension. For instance, the dimension of a vector space is the size of any its basis (I recall that all bases of a vector space have the same size, see, for instance, [Lan,III, $\S$5]]). But note that the dimension depends on $F$. For instance, $\mathbb R$ has dimension $1$ as a vector space over the field $\mathbb R$, but dimension $\mathfrak c$ as a vector space over the field $\mathbb Q$. Fortunately, given the scalars, we are already pointed over which field we have to consider them. Namely, we are considering the field of scalars $F$ as a vector space over $F$. Then any nonzero element of $F$ constitutes a basis of $F$ over $F$, so the dimension of $F$ over $F$ is one.

References

[Lan] Serge Lang, Algebra, Addison-Wesley, Reading, Mass., 1965 (Russian translation, Moskow, 1968).

[M] Mathematical encyclopedia, vol. 4, Soviet encyclopedia, Moskow, 1984, in Russian.

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  • $\begingroup$ we are considering the field of scalars F as a vector space over F. But now what were "scalars" are no longer scalars but vectors? $\endgroup$
    – user1210203
    Sep 24, 2023 at 6:24
  • $\begingroup$ @user103496 Speaking informally, when we consider $F$ as a vector space over $F$ we have two points of view on $F$. From the first of them, elements of $F$ are vectors, that is elements of the vector space $V$, $V=F$ (so $V$ is a one-dimensional vector space over $F$, for instance, with the basis $\{1\}$). From the second point of view, $F$ is the field of scalars for the vector space $V$. $\endgroup$ Sep 24, 2023 at 6:36
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Dimension is a concept that doesn't apply that well to scalars. As Ninad Munshi said in their reply, dimension refers to the vector space in which the matrix/vector is embedded. Vector spaces have a "dimension"; vectors have "rank" (which is basically just the number of elements in the vector; e.g. [3,2] has rank 2 while [7,1,10] has rank 3, etc). Scalars don't really have either.

In other words, a scalar is not simply a rank 1 vector or a rank 1 matrix. Scalars are a different ingredient in the logic of linear algebra. They can be taken from a different space than the vector space (called fields). Often we take our scalars from the Real numbers, which is also where the elements of our vectors/matrices come from in many situations; but this is just coincidence. We could decide "in this scenario, we'll use only Integers to build our vectors, but use Real numbers as our scalars" -- and so on.

To summarize: scalars are different creatures from vectors; they come from a different realm. So, the concepts we use to describe vectors don't really apply to scalars.

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    $\begingroup$ This is just wrong from reading the examples you used. Everything has a rank. The vector $(3,2)$ lives in a 2 dimensional space, but because vectors arrange information in a sequential list, it is a rank 1 tensor. Rank refers to how the information is organized, not how big the information is. $\endgroup$ Aug 28, 2023 at 8:41
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If your scalars are single real numbers, $c \in \mathbb R $ then they could be considered 1 dimensional. Contrast a one dimensional number line, $\mathbb R$ occasionally notated $\mathbb R ^1$, with a 2 dimensional xy plane Or even a 3 dimensional xyz space, $\mathbb R^3 $. Be aware, the symbol. $\mathbb R$, stands for the set of real numbers.

Note: There is no universal agreement on this as the other answers point out.

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