Why do we call Vector Spaces Vector Spaces and not Linear Spaces?

Question

When thinking about vector spaces, the image most people will have in mind is that of arrows in euclidean space. Indeed, in common dictionaries we may find definitions such as:

• Cambridge dictionary: "a quantity such as velocity that has both size and direction"
• Oxford dictionary: "​(mathematics) a quantity that has both size and direction"
• Merriam-Webster: "a quantity that has magnitude and direction and that is commonly represented by a directed line segment whose length represents the magnitude and whose orientation in space represents the direction"
• Duden: "Größe, die als ein in bestimmter Richtung mit bestimmter Länge verlaufender Pfeil dargestellt wird und die durch verschiedene Angaben (Richtung, Betrag) festgelegt werden kann"

Yet, all we find when looking in a mathematics text is "an element of a vector space". And subsequently, after looking up the definition of a vector space we are shocked to find out that an element of a vector space neither needs to have a direction, nor a magnitude.

Indeed, if we seek to be able to perform the standard operations we would expect to be able to do with vectors, such as:

• measuring the length of vectors
• measuring the angle between pairs of vectors
• measuring the area spanned by a pair of vectors
• measuring the volume of a paralellepiped spanned by multiple vectors

We quickly realise that we will at least need an inner product (or pre-Hilbert) space.

My question: Where did this disconnect originate? Why are we calling mathematical vector spaces vector spaces, even though it goes against common understanding of what a vector entails?

Opinionated: Shouldn't we be calling vector spaces linear spaces and inner product spaces vector spaces instead?

Answer 1

I suppose because the idea of having a vector space is the generalization of the vector euclidean space. Many abstract structures are the attempt to generalize some particular structure that behaves "well", and thus see what other sets also behave "well" in the area that we are interested in studying.

Answer 2

There is at least one contemporary textbook that does precisely this, for exactly the reasons you suggest: Linear Algebra with Applications by Otto Bretscher. In this book (which is used at my University for the standard introductory linear algebra course) abstract vector spaces are called "linear spaces", and the phrase "vector space" is reserved exclusively for $\mathbb R^n$. When the term is first introduced (p. 167), a footnote explains:

The term "vector space" is more commonly used in English (but it's espace linéaire in French). We prefer the term "linear space" to avoid the confusion that some students experience with the term "vector" in this abstract sense.

As for why this isn't more common, I doubt very much that this is an answerable question. Language and usage tend to evolve in haphazard ways, and once a custom is established it is hard to dislodge.

Answer 3

If you must do linguistic nitpicking, then one can argue that "vector space" is a better match to the abstract notion than "linear space". Linearity ultimately derives from the geometric notion of "line" (in an affine space) which is quite some distance removed from the property that makes a set into a vector space; in the modern mathematical sense the adjective "linear" is applied to maps and transformations, and refers to compatibility with addition and scalar multiplication (or more conceptually with linear combinations). In this setting a plausible name for vector spaces would be "linearity spaces": the spaces themselves are not in any sense linear, but they have the structure (namely linear combinations) necessary to be able to talk about linearity of maps.

By contrast "vector" derives from the Latin verb "vehere" (to transport, a stem also found in "vehicle"), so the notion of vector suggests an object that produces a displacement. It is true that a displacement can usually be described by its direction and quantity (distance of displacement), in which case two displacements need to have both the same direction and quantity to be considered equal. But that does not mean direction and magnitude are the defining characteristics of displacement; indeed one can for instance easily consider simultaneous displacements in time and space, for which displacements the notions of direction and magnitude have no obvious meaning. I would therefore consider the dictionary descriptions more as clumsy attempts to capture a mathematical notion in layman's language, than as a specification for the notion. More fundamental aspects of displacements are that they can be combined (added) or multiplied by a factor; as such they do mesh quite nicely with the fundamental operations encompassed in the notion of a vector space.

As for the "standard operations we would expect to be able to do with vectors" you mention, they just reflect your preconceptions; they are in no way directly associated with the notion of vector. They are indeed things one can do in Euclidean vector spaces, and for lack of imagination we do not have a separate terms for elements of such spaces; with many vector spaces arising in physics being in fact equipped with an inner product, one does often encounter such measurements being associated with vectors, but that does not imply they are defined for general vector spaces.