TITLE: Covectors and Vectors
QUESTION [18 upvotes]: I have a general question about vector/covectors: 
Background. A vector (for our purposes) is a physical object in each basis of $\mathbb{R}^3$ represented by three numbers such that these numbers obey certain transformation rules when we change the basis. Let $\textbf{x}$ be an arbitrary vector and $\textbf{e}_1, \textbf{e}_2, \textbf{e}_3$ and $\tilde{\textbf{e}}_1, \tilde{\textbf{e}}_2, \tilde{\textbf{e}}_3$ be two bases. These transformation/inverse transformation rules are the following: 
$$\tilde{x}^{j} = \sum_{j=1}^{3} T_{i}^{j} x^i$$ and $$x^{j} = \sum_{i=1}^{3} S_{i}^{j} \tilde{x}^i$$

Question. Vectors satisfy the above properties. Now if I imagine some other set of objects that satisfy the above properties why do we call them covectors? What are covectors and how are they different from vectors if they satisfy the same properties?

REPLY [4 votes]: The (physical) distinction between vectors and covectors is related to frame of reference changes.
Since you asked about $3$-vectors, I will refer just to spatial frame of reference changes, but a similar rule holds for 4-vectors and space-time frame of reference changes.
Let $O$, $O'$ be two spatial frames of reference, $(x_1, x_2, x_3)$ are the coordinates of an event in $O$, $(x'_1, x'_2, x'_3)$ the coordinates of the same event in $O'$; they are related by
$$
x'_i = \sum_{j=1}^{3} T_{ij} x_j + k_i
$$
Let $A$ be an observable completely defined by $3$ numbers, $(A_1, A_2, A_3)$ are the values of those numbers in $O$, $(A'_1, A'_2, A'_3)$ the values in $O'$.
Iff the following equality holds
$$
A'_i = \sum_{j=1}^{3} T_{ij} A_j
$$
$A$ is said to be a vector.
Otherwise, iff the following one holds
$$
A'_i = \sum_{j=1}^{3} (T^{-1})_{ji} A_j
$$
$A$ is said to be a covector.
In other words, vectors transform with the same matrix $T$ as space coordinates and covectors with the matrix $(T^{-1})^{tr}$.
Usually, one uses orthogonal coordinate systems with same length unit of measurement, so $T$ is just an $O(3)$ matrix. Under that condition, we have
$$
T = (T^{-1})^{tr} 
$$
and covectors are undistinguishable from vectors. This fact explains why we rarely speak of covectors in classical mechanics.
The situation radically changes with space-time frames of reference in special relativity.
In that case, $T$ is an $O(1, 3)$ matrix, $T$ is (in general) different from $(T^{-1})^{tr}$ and, vectors and covectors are very distinct.
As a final remark, note that there are "vector-like" (classical) physical quantities that are neither vectors nor covectors.
The angular momentum of  a point mass is defined by
$$
\mathbf l = m \mathbf r\times \mathbf v 
$$
where $m$, $\mathbf r$ and $\mathbf v$ are, respectively, mass, position and speed of the body.
If $(l_1, l_2, l_3)$ are the components of $\mathbf l$ in $O$, $(l'_1, l'_2, l'_3)$ the components in $O'$ and $T$ is a rotation (that is $T\in SO(3)$) then it's easy to verify that
$$
l'_i = \sum_{j=1}^3 T_{ij} l_j
$$
So $\mathbf l$ behaves like a vector with respect to rotations.
But if the transformation from $O$ to $O'$ is
$$
x'_i = -x_i
$$
then we have
$$
l'_i = l_i
$$
Though often $\mathbf l$ is referred to as a vector, it is neither a vector nor a covector. It is more appropriate calling it pseudovector.