  Every m by n matrix A with rank r has exactly r independent rows and r
independent columns. The fact has become the most fundamental theorem in linear
algebra such that we may favor it in an unconscious way. The sole aim of this
paper is to give a self-contained introduction to concepts and mathematical
tools for the rank of a matrix in order to seamlessly introduce how it works in
applied linear algebra. However, we clearly realize our inability to cover all
the useful and interesting results concerning this topic and given the paucity
of scope to present this discussion, e.g., a proof via the injective linear
map. We refer the reader to literature in the field of linear algebra for a
more detailed introduction to the related fields.
