When studying physical problems, we are used to describing vectors in a coordinate system with a set of numbers. However, it is arbitrary to introduce an additional coordinate system, and different coordinate system choices will lead to different difficulties, or we sometimes need to analyze the same system in different coordinate systems to study the symmetry of the system, so we need to change the selected coordinate system frequently. When changing the coordinate system, it is obvious that the number of this set of description vectors will change, while some quantities will remain unchanged (for example, the inner product of vectors is only related to the length and angle of vectors). Since physical phenomena should have nothing to do with the coordinate system that describes them, those things that remain unchanged when changing the coordinate system are more worthy of attention, such as the length and angle of vectors, in other words, the inner product. If we only study the rotation of coordinate system in rectangular coordinate system, we can use an orthogonal matrix to describe the relationship between the components of the vector. From the properties of orthogonal matrix, it can be easily concluded that the inner product of vector is invariant before and after coordinate system rotation. But if we want to study the expansion of coordinate system and non-orthogonal coordinate system, then the matrix describing vector transformation becomes non-orthogonal matrix. If we use the definition of the inner product of orthogonal matrix to calculate, we will get a number related to the reference system. As mentioned earlier, physicists don't care about the objects on which this coordinate system depends. At this time, we need to describe a vector with two sets of numbers, one of which changes in a normal way, and the other changes in the form of the inverse matrix of the transformation matrix (instead of the transposed matrix in the rectangular coordinate system). When two vectors do inner product, you can still get a coordinate-independent number. We call these two vectors covariant vectors and find the inverse quantity, and there will be another set of coordinate frames that change in the form of the inverse matrix of the original coordinate frame. These two sets of coordinate systems are also called covariant coordinate system and inverse coordinate system. At the same time, each coordinate frame will have a matrix, which can transform covariant vectors and inverse quantities into each other, which is called measurement. The inverse of a matrix is mutual, so covariation and inversion are also mutual. Which one is called covariation and which one is called inversion, to a great extent, comes from history and convention. Covariance and inversion are just two components of a vector or tensor. They can be related by metric tensor.