Matrix Rank

Row Echelon Form: diagonal 1s from top left to bottom right with bottom part of diagonal being zeros and top part being any number. (http://mathworld.wolfram.com/EchelonForm.html)

Rank of Matrix: how many non zero rows there are after echelon form has been calculated.

Ax = 0 can be easily solved using the reduced REF. Can find x in terms of arbitrary values(eg: a & b) which means there is an infinite number of solutions of x. Called a solution space.

This is a general solution.

In matrix m x n, m is columns and n is rows. r = rank of matrix.

If r < n,

When Ax = b (a non zero vector), the solution is the particular solution to Ax = b and the general solution of Ax = 0. This is x = p + v.

p is the particular solution to Ax = b, v is the general solution to Ax = 0.

Suppose matrix A is m x n with rank r

if r = n, there exists a unique solution

if r < n, there exists an infinite number of solutions