# When is a vector a tensor? Is invariance under affine transformation required?

Several resources, such as this (pages 11 & 15) and this, state that position vectors are not rank one tensors while displacement vectors are, since only displacement vectors are invariant under a translation of the origin. For example:

The key attribute of a tensor is that its representations in different coordinate systems depend only on the relative orientations and scales of the coordinate axes at that point, not on the absolute values of the coordinates. This is why the absolute position vector pointing from the origin to a particular object in space is not a tensor, because the components of its representation depend on the absolute values of the coordinates. In contrast, the coordinate differentials transform based solely on local information.

This leaves me confused, since a translation of the basis seems to be equivalent to an affine transformation. This seems to claim that a property of tensors is invariance under affine transformations.

However, the "multidimensional arrays" definition of tensors given on Wikipedia seems to be in terms of only linear transformations. Furthermore, an affine transformation on a linear vector space would seem to preserve only the affine structure, but not the linear structure, so that position vectors are removed by the transformation anyway. In fact, if the set of all position vectors from the origin to points in $\mathbb{R}^n$ forms a vector space, while the set of all displacement vectors between points in $\mathbb{R}^n$ forms an affine space, then it would seem no vectors could ever be tensors. Only elements of a nonlinear affine space could be invariant under affine transformations, and therefore tensors.

This makes it seem unlikely that invariance under translations of the coordinate system is required - only invariance under linear transformations of the basis. Then position vectors, as well as all other vectors, should in fact be tensors. Is this correct, or am I misunderstanding the concepts somewhere?

Defining geometric objects by their coordinate transformation laws is common in physics, particularly older physics texts, but it's not so common in math these days. To most modern geometers, the difference between positions and displacements is a difference in algebraic structure: the operations you can do with positions are different from the ones you can do with displacements. I'll answer your question from that perspective.

Imagine two-dimensional space as an endless, unmarked sheet of paper—no origin, no grid lines, just a blank. Although the paper doesn't come with any predefined landmarks, you can make some by drawing dots on it with a pencil. Those dots are positions.

Wandering around, you occasionally come across arrows lying on the paper. You can slide the arrows around, but you can't change their lengths or their orientations. Those arrows are displacements.

There are a bunch of operations you can do with dots and arrows. (I listed them all for completeness, but you can skip to the end if you get tired of reading them.)

• Given a dot $P$ and an arrow $v$, drag $v$ so its tail lies on $P$. The tip of $v$ points to a new dot, which I'll call $P + v$. This defines an operation, "translation," that takes in a dot and an arrow and spits out a new dot.

• For any two dots $P$ and $Q$, there's exactly one arrow that runs from $P$ to $Q$. In other words, there's exactly one arrow $v$ such that $P + v = Q$. I'll call that arrow $Q - P$. This defines an operation, "subtraction," that takes in two dots and spits out an arrow.

• Given two arrows $v$ and $w$, drag $w$ so its tail lies at the tip of $v$. There's exactly one arrow that runs from the tail of $v$ to the tip of $w$, which I'll call $v + w$. This defines an operation, "addition," that takes in two arrows and spits out another arrow. Keep in mind that addition is different from translation, even though we use the same symbol for both of them.

• We can flip an arrow $v$ so it's pointing the opposite direction, keeping its length the same. I'll call the resulting arrow $-v$. This defines an operation, "negation," that takes in an arrow and spits out another arrow.

• Given an arrow $v$ and a number $\lambda \ge 0$, we can scale the length of $v$ by a factor of $\lambda$, keeping its direction the same. I'll call the resulting arrow $\lambda v$. This defines an operation, "multiplying," that takes in an arrow and a non-negative number and spits out an arrow.

• You can combine multiplying and negation into a single operation, "multiplication," that takes in an arrow and a number—negative or not—and spits out an arrow.

For these operations to make sense, you have to keep track of the difference between dots and arrows. There's no obvious way, for example, to add two dots, or to multiply a dot by a number. That's the difference between positions and displacements: the operations you can naturally do with them are different.

A space with the kind of algebraic structure described above is called an affine space.

• Those are probable the most lucid explanations of vector operations I've come across while learning the basics of linear algebra. I wish I could find similarly intuitive explanations for linear (or multilinear) transformations like rotation, etc. – Bert Zangle Dec 12 '16 at 17:29

See my explanation of the difference between translations (functions) and pictures (arrows) of their effects