Rank From Row-echelon Forms
A common approach to finding the rank of a matrix is to reduce it to a simpler form, generally row-echelon form by row operations. Row operations do not change the row space (hence do not change the row rank), and, being invertible, map the column space to an isomorphic space (hence do not change the column rank). Once in row-echelon form, the rank is clearly the same for both row rank and column rank, and equals the number of pivots (or basic columns) and also the number of non-zero rows, say p; further, the column space has been mapped to which has dimension p.
A potentially easier way to identify a matrices' rank is to use elementary row operations to put the matrix in reduced row-echelon form and simply count the number of non-zero rows in the matrix. Below is an example of this process.
Matrix A can be put in reduced row-echelon form by using the following elementary row operations:
By looking at the final matrix (reduced row-echelon form) one could see that the first non-zero entry in both and is a 1. Therefore the rank of matrix A is 2.
Read more about this topic: Rank (linear Algebra)
Famous quotes containing the words rank and/or forms:
“West of this place, down in the neighbor bottom,
The rank of osiers by the murmuring stream
Left on your right hand brings you to the place.”
—William Shakespeare (1564–1616)
“The Dada object reflected an ironic posture before the consecrated forms of art. The surrealist object differs significantly in this respect. It stands for a mysterious relationship with the outer world established by man’s sensibility in a way that involves concrete forms in projecting the artist’s inner model.”
—J.H. Matthews. “Object Lessons,” The Imagery of Surrealism, Syracuse University Press (1977)