ABSTRACT
A vector space is an algebraic abstraction of the set of vectors (arrows) in the plane. A vector space over a field F is an additive abelian group V whose elements can be multiplied by elements of F. The elements of V are called vectors, while the elements of F, in this context, are called scalars. If the number of elements in a basis of a vector space is finite, it is called the dimension of the vector space, and the vector space is finite-dimensional. In order to compare the row spaces of matrices it is convenient to put the matrices in a standard form. This is done by means of elementary row operations. A matrix is said to be in row-echelon form if each nonzero row precedes each row of zeros. A matrix in row-echelon form is in reduced row-echelon form if the first nonzero entry in any nonzero row is the only nonzero element in its column.
