ABSTRACT
With the help of vector notation any m × n system of linear equations may be written as a single equation. The way of writing the equations puts the problem in a different light: to solve the vector equation is to express the vector on the right as a linear combination of the two vectors on the left. The traditional process of elimination consists in adding multiples of one equation to another. In other words, mathematicians modify the system of equations in a way which does not affect the solutions. All such ways of operating on such a system can be obtained as a succession of the several 'elementary' operations. The chapter focuses on the question about the invariance of the number of elements in a basis of a vector space.
