Scalar-valued function definition

Question

A scalar-valued function is defined to be a function with a single number as its output. What is the logical origins of giving this type of function the name “scalar-valued”. It seems like a misleading name because a scalar is supposed to only have magnitude and no direction but a single number can be either positive or negative. Thus, what is “scalar” referring to in “scalar-valued”?

Answer 1

Short Answer: Your definitions are inexact, and that is leading you to confusion. A scalar is an element of the field over which a vector space is defined. Historically, the term is related to the geometric interpretation of a vector as an arrow, and a scalar as a thing which can change the length (or "scale") an arrow, but you should not be led down the garden path by familiar English terms in a mathematical context, where they may gain technical meaning.

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Longer Answer: We should be careful in mathematics about using the right definitions. I am not sure that I have ever seen a scalar-valued function defined as a function which has "a single number as its output." Your definition of a scalar as a thing(?) with only "magnitude and no direction" is similarly suspicious. While these are often workable definitions in an elementary setting where the goal is to teach things in a broad-strokes manner with an eye toward computation, these are not good definitions for addressing your question.

To improve the situation, let's start with some more rigorous definitions:

Definition: A field is a triple $k = (k,+,\cdot)$, consisting of a set $k$ and two binary operations $+$ and $\cdot$, which satisfy certain axioms (basically, addition and multiplication are both associative and commutative, multiplication distributes over addition, and there are distinguished additive and multiplicative identities).

Some basic examples of fields are $\mathbb{Q}$ (the rationals), $\mathbb{R}$ (the reals), and $\mathbb{C}$ (the complex numbers) — indeed the field axioms basically generalized the usual addition and multiplication so that we can study those structures without having to worry about the details of these fields.

Definition: A vector space is a quadruple $V = (V,+,k,\cdot)$, where $V$ is a set, $+$ is a binary operation which acts on elements of $V$, $k$ is a field, and $\cdot$ is an operation which takes an element of $k$ and an element of $V$ as input, and outputs an element of $V$. These operations are subject to axioms, which basically say that the operations behave the way we want them to.

In this context, the elements of $V$ are called vectors, and the elements of $k$ are called scalars (and $k$ is the field of scalars, or the base field). We will also often say that $V$ is a vector space "over" the field $k$.

A standard example of a vector space is $\mathbb{R}^n$, which consists of a set $$ V = \mathbb{R}^n = \{ \langle x_1, x_2, \dotsc, x_n \rangle : x_j\in\mathbb{R}\forall j\in\{1,2,\dotsc,n\} \}, $$ addition of vectors is defined element-wise, the field of scalars is $\mathbb{R}$, and the multiplication operation (referred to as scalar multiplication) "distributes" over the terms of a vector, that is, if $\alpha \in \mathbb{R}$ and $v = \langle x_1, x_2, \dotsc, x_n \rangle$ is a vector, then $$ \alpha v = \alpha \langle x_1, x_2, \dotsc, x_n \rangle = \langle \alpha x_1, \alpha x_2, \dotsc, \alpha x_n \rangle. $$ In this vector space, the vectors and scalars have (or can be interpreted as having) geometric meaning: a vector is an abstract arrow of some length with points in some direction, and scalar multiplication serves to change the length of an arrow (and, possibly, to reverse its orientation)—here, the term "scalar" is slightly less abstract.

Finally, we have what we need to define a "scalar-valued function":

Definition: A function $$ f : V \to k $$ is called a scalar-valued function.

Put simply, a scalar-valued function is a function which takes a vector (one of these weird, abstractly defined objects) as input, and spits out an element of the base field as output. As noted above, the "scaling" here is something of a historical anachronism, or an example of a standard English word which has taken on technical meaning in mathematics (other such words include "normal", which has a ton of technical uses in mathematics; "exact", which has special meaning in algebra; and "smooth" is defined in terms of differentiability). You should be careful not to get misled by vernacular uses of technical terms.