What is the number of dimensions of a scalar?

Question

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.

Answer 1

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.

Answer 2

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.

Answer 3

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.