What exactly is a scalar?

Question

Particularly, referring to scalars I am only implying to individual numbers, for this instance, let's take the set R(real numbers).

These numbers do lie of the real line, i.e. the x axis, which is perpendicular to the y-axis, together form the cartesian plane. So, how are they scalar?

For instance, the number 5 can very well be represented by (5, 0) in the plane, which could also imply that it is a position vector.

If so, let u & v be two vectors in the cartesian plane, then what would the dot product u.v imply? How can the value of it be interpreted? It's obviously not just a random number formed by multiplication of two vectors, what is it that the value of the dot product is so significant?

Answer 1

There are two parts to this question: What is a scalar?, and Can we represent them as vectors? I think these both stem from the same confusion.

In math, we don't usually think about things by what they are, but rather by how they act. So, if we are thinking about the vector space $\mathbb{R}^{2}$ (over $\mathbb{R}$), it seems like we can identify the set $\{ (x, 0) : x \in \mathbb{R} \}$ with the real numbers. They act somewhat similarly, but they have an essential difference: multiplication.

To get what I mean by that, think about how we define vector spaces. Ignore the dot product for now, since it is not central to vector spaces, but rather something we tack on at a later point (when talking about Inner Product Spaces).

A vector space has two parts to it: the vectors, and the scalars. The vectors are simply things that we can add together, with some rules to make it work nicely. For scalars, the essential operations are multiplication and scaling (we require addition as well, but that is somewhat irrelevant to their purpose). In essence, scalars are built around the idea that we use them to scale vectors, and being able to multiply them is actually a natural consequence of that (for a vector $u$, and scalars $a, b$, it is nice express scaling by both as scaling by their product: $a \cdot (b \cdot u) = (ab) \cdot u$).

More precisely, we can think of scalars by how they act on vectors: given a scalar $a$, we can think about the function $u \mapsto a \cdot u$ (for vectors $u$) which represents scaling by $a$. Again, with this it is tempting to take a particular vector $u$ and identify each scalar $a$ with $a \cdot u$. They work the same additively, but they lose the essential ability to scale (in other words, multiplication). Furthermore, this is not a unique representation, since we could choose any vector $u$. It is interesting, but it is not all that helpful.

Now, this is particularly tempting with $\mathbb{R}^{2}$ because the identification with the $x$-axis seems natural, but one of the central points of linear algebra is that it really is not. Basically, the particular coordinate systems that we choose are not really a meaningful part of the structure, and so it is just as natural to choose any one to represent $\mathbb{R}$.

This is primarily true since we are talking about vector spaces, however. If we were to talk about the complex numbers $\mathbb{C}$, it is a different story. Basically, $\mathbb{C}$ is a vector space over $\mathbb{R}$, and looks exactly like $\mathbb{R}^{2}$, but it has an additional structure: multiplication. Since the complex numbers work like the real numbers (we have addition and multiplication), it is possibly to uniquely represent $\mathbb{R}$ a subset of $\mathbb{C}$. We usually write complex numbers in the form $a + bi$ for real numbers $a, b$, which suggests using $a + 0i$ to represent $a$. In fact, this is precisely the right way to do so to get both addition and multiplication to work right.

Lastly, on the dot product, it really is completely different than it first looks. It is not really multiplication on vectors since the result is a scalar. More precisely, it is not multiplication in the way we have for $\mathbb{R}$ and $\mathbb{C}$ (i.e. for fields). There are all sorts of things we can do in $\mathbb{R}$ (like division) that do not make much sense in $\mathbb{R}^{n}$, even with the dot product.

Really, the most important thing to remember when studying vector spaces (and not more complicated structures like inner product spaces) is that we want everything to be independent of the coordinate system (i.e. basis) that we choose. The dot product is problematic because it makes the standard coordinate system special. Identifying the scalars with a particular axis is problematic because it makes that axis special. Vector addition is good since it is coordinate independent (etc).

Answer 2

There's no such thing as a scalar. But there's a notion of field, and if $k$ is a field, we can speak of the vector spaces over $k$. Sometimes, we've fixed a field $k$, and we're studying vector spaces over it. In this case, we sometimes write "let $a$ denote a scalar" instead of the more formal "let $a$ denote an element of $k$." But that's all it is - a terminological gimmick, no more.

And you're completely correct that $k$ can itself be viewed as a vector space over $k$. This allows us to can say things like "every scalar can be viewed as a vector in a $1$-dimensional vector space" etc.

Answer 3

Here's part of the definition of a vector space (taken from here): "A vector space over a field F is a set V together with two operations that satisfy the eight axioms listed below. Elements of V are commonly called vectors. Elements of F are commonly called scalars."

The two operations are usually called vector addition and scalar multiplication. This is also a generalization of, what is known as, a module. When the set of scalars is not 'only' a ring but actually a field, then it is called a vector space.

The scalars are possibly called so because they 'scale' the vectors, but it's of course just a name. For example, when you multiply an 'ordinary' vector, say $(1,2)$ in $\mathbb{R}^2$, with the scalar $3$, you get $(3,6)$, which has the same direction but a greater magnitude.

So in a vector space, you always have a set of objects called vectors and a set of scalars, obeying a list of axioms. The vector spaces are often referred to as a $K$-vector space, where $K$ is the scalar field. We also just say a real vector space about an $\mathbb{R}$-vector space.

Any field, such as the real numbers, could then be considered a vector space over itself. This would then be a set of vectors ($\mathbb{R}$) with a set of scalars ($\mathbb{R}$), where vector addition is the ordinary addition in $\mathbb{R}$ and scalar multiplication is the ordinary multiplication in $\mathbb{R}$. So the real numbers could of course be called scalars in that sense.