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==Introduction==
In physics, a vector typically arises as the outcome of a measurement or series of measurements, and is represented as a list (or [[tuple]]) of numbers such as

:<math>(v_1,v_2,v_3) .</math>

The numbers in the list depend on the choice of [[coordinate system]].  For instance, if the vector represents position with respect to an observer ([[position vector]]), then the coordinate system may be obtained from a system of rigid rods, or reference axes, along which the components ''v''<sub>1</sub>, ''v''<sub>2</sub>, and ''v''<sub>3</sub> are measured.  For a vector to represent a geometric object, it must be possible to describe how it looks in any other coordinate system.  That is to say, the components of the vectors will ''transform'' in a certain way in passing from one coordinate system to another.

A simple illustrative case is that of a [[Euclidean vector]].  For a vector, once a set of basis vectors has been defined, then the components of that vector will always vary ''opposite'' to that of the basis vectors.  That vector is therefore defined as a ''contravariant'' tensor.  Take a standard position vector for example. By changing the scale of the reference axes from meters to centimeters (that is, ''dividing'' the scale of the reference axes by 100, so that the basis vectors now are <math>.01</math> meters long), the components of the measured [[Position (geometry)|position]] [[Vector (mathematics and physics)|vector]] are ''multiplied'' by 100.  A vector's components change scale ''inversely'' to changes in scale to the reference axes, and consequently a vector is called a ''contravariant'' tensor.

A ''vector'', which is an example of a ''contravariant'' tensor, has components that transform inversely to the transformation of the reference axes, (with example transformations including [[rotation (mathematics)|rotation]] and [[Dilation (metric space)|dilation]]). [[Active and passive transformation#Passive transformation|The vector itself does not change under these operations]]; instead, the components of the vector change in a way that cancels the change in the spatial axes.  In other words, if the reference axes were rotated in one direction, the component representation of the vector would rotate in exactly the opposite way. Similarly, if the reference axes were stretched in one direction, the components of the vector, would reduce in an exactly compensating way.  Mathematically, if the coordinate system undergoes a transformation described by an <math>n\times n</math> [[invertible matrix]] ''M'', so that the basis vectors transform according to <math>\begin{bmatrix} \mathbf{e}_1^\prime\  \mathbf{e}_2^\prime\  ... \ \mathbf{e}_n^\prime\end{bmatrix}=\begin{bmatrix} \mathbf{e}_1\  \mathbf{e}_2\  ... \ \mathbf{e}_n\end{bmatrix}M</math>, then the components of a vector '''v''' in the original basis ( <math>v^i</math> ) must be similarly transformed via <math>\begin{bmatrix} v^1{^\prime} \\ v^2{^\prime} \\ ... \\ v^n{^\prime} \end{bmatrix}=M^{-1}\begin{bmatrix} v^1 \\ v^2 \\ ... \\ v^n \end{bmatrix}</math>.  The components of a ''vector'' are often represented arranged in a column.

By contrast, a ''covector'' has components that transform like the reference axes.  It lives in the dual vector space, and represents a linear map from vectors to scalars. The dot product operator involving vectors is a good example of a covector.  To illustrate, assume we have a covector defined as  <math>\mathbf{v}\ \cdot</math> , where <math>\mathbf{v}</math> is a vector.  The components of this covector in some arbitrary basis are  <math>\begin{bmatrix} \mathbf{v}\cdot\mathbf{e}_1 &  \mathbf{v}\cdot\mathbf{e}_2 & ... & \mathbf{v}\cdot\mathbf{e}_n  \end{bmatrix}</math>, with <math>\begin{bmatrix} \mathbf{e}_1\  \mathbf{e}_2\  ... \ \mathbf{e}_n\end{bmatrix}</math> being the basis vectors in the corresponding vector space. (This can be derived by noting that we want to get the correct answer for the dot product operation when multiplying by an arbitrary vector <math>\mathbf{w}</math> , with components <math>\begin{bmatrix} w^1 \\ w^2 \\ ... \\ w^n \end{bmatrix}</math>).  The covariance of these covector components is then seen by noting that if a transformation described by an <math>n\times n</math> [[invertible matrix]] ''M'' were to be applied to the basis vectors in the corresponding vector space, <math>\begin{bmatrix} \mathbf{e}_1^\prime\  \mathbf{e}_2^\prime\  ... \ \mathbf{e}_n^\prime\end{bmatrix}=\begin{bmatrix} \mathbf{e}_1\  \mathbf{e}_2\  ... \ \mathbf{e}_n\end{bmatrix}M</math>, then the components of the covector <math>\mathbf{v}\ \cdot</math> will transform with the same matrix <math>M</math>, namely, <math>\begin{bmatrix} \mathbf{v}\cdot\mathbf{e}_1^\prime &  \mathbf{v}\cdot\mathbf{e}_2^\prime & ... & \mathbf{v}\cdot\mathbf{e}_n^\prime  \end{bmatrix}=\begin{bmatrix} \mathbf{v}\cdot\mathbf{e}_1 &  \mathbf{v}\cdot\mathbf{e}_2 & ... & \mathbf{v}\cdot\mathbf{e}_n  \end{bmatrix}M</math>.  The components of a ''covector'' are often represented arranged in a row.

A third concept related to covariance and contravariance is [[Invariant (physics)|invariance]].  A [[scalar (physics)|scalar]] (also called type-0 or rank-0 tensor) is an object that does not vary with the change in basis.  An example of a physical [[observable]] that is a scalar is the [[mass]] of a particle.  The single, scalar value of mass is independent to changes in basis vectors and consequently is called ''invariant''. The magnitude of a vector (such as [[Euclidean distance|distance]]) is another example of an invariant, because it remains fixed even if geometrical vector components vary.  (For example, for a position vector of length <math>3</math> meters, if all [[Cartesian coordinate system|Cartesian]] basis vectors are changed from <math>1</math> meters in length to <math>.01</math> meters in length, the length of the position vector remains unchanged at <math>3</math> meters, although the vector components will all increase by a factor of <math>100</math>).  The scalar product of a vector and a covector is invariant, because one has components that vary with the base change, and the other has components that vary oppositely, and the two effects cancel out.  One thus says that covectors are ''dual'' to vectors.

Thus, to summarize:
* A vector or [[tangent vector]], has components that ''contra-vary'' with a change of basis to compensate. That is, the matrix that transforms the vector components must be the inverse of the matrix that transforms the basis vectors. The components of vectors (as opposed to those of covectors) are said to be '''contravariant'''. In [[Einstein notation]] (implicit summation over repeated index), contravariant components are denoted with ''upper indices'' as in
*:<math>\mathbf{v} = v^i \mathbf{e}_i</math>
* A covector or [[cotangent vector]] has components that ''co-vary'' with a change of basis in the corresponding (initial) vector space. That is, the components must be transformed by the same matrix as the change of basis matrix in the corresponding (initial) vector space. The components of covectors (as opposed to those of vectors) are said to be '''covariant'''. In [[Einstein notation]], covariant components are denoted with ''lower indices'' as in
*:<math>\mathbf{w} = w_i \mathbf{e}^i.</math>
* The scalar product of a vector and covector is the scalar <math>v^iw_i</math>, which is invariant.  It is the duality pairing of vectors and covectors.