In linear algebra, matrixes are used to change basis, to translate between spaces. It must be square, therefore (why?). However, there are known transformations between infinite conutable and uncountable vectors. Obviously, their dimentions do not match and matrix is uncountable x countable rather than square.
When a number of vectors (columns) than the length of a vector in column space, there are zero rows in eschelon expantion and solution not always exists.