What is the rank of a matrix?

If matrix A has an R-level sub-formula not zero, all r + 1 sub-formulas (if there is R + 1 sub-formula) are equal to zero, it can be introduced that all the subformulas of higher order of a are zero, so r is the highest order of non-zero subformulas of A. This order is called the rank of. This is the definition of the rank of the matrix in the book. What is the more essential meaning of the rank of the matrix, or what is expressed in the linear geometric space?

When it comes to the rank of the matrix, we have to mention the rank of the vector group, that is, the number of vectors contained in the extremely large irrelevant group of the vector group, A large unrelated group is a set of linear unrelated vectors in a vector group, and any vectors in the vector group can be listed in the linear form of this set of linear irrelevant vectors.

A matrix is nothing more than a set of vectors. It is a linear space composed of these vectors, and the rank of the matrix is equal to the row rank and column rank, that is to say, the rank of the matrix is equal to the rank of the vector group that forms the matrix. Each vector in this vector group can be listed in a linear form by a large unrelated vector group in this group. That is to say, there is such a linear space, all vectors in a space can be expressed by a specific set of vectors. This set of specific vectors determines such a linear space. This set of specific vectors is the extremely large irrelevant group in the vector group, the number of irrelevant groups is the dimension of the vector space.

For example, for a two-dimensional plane space that we are familiar with, the two vectors representing the coordinate axes constitute a very large irrelevant group in the space, any vector in the space can be listed by the two vectors in linear form. The rank of the space is the dimension of the space.
The above is only a rough discussion. If you have other ideas, please correct them.