# Solving Ax = 0, Pivot Variables, and Special Solutions

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Lesson Summary:

In this lesson, we learn how to solve Ax=0 using elimination to get a matrix into reduced row echelon form. The algorithm involves finding the pivot and free variables, assigning values to the free variables, and solving for the pivot variables. The number of free variables is equal to the number of columns minus the rank of the matrix. The null space contains all combinations of the special solutions, where there is one special solution for each free variable. Finally, the matrix is transformed into reduced row echelon form by using elimination upwards to get zeros above the pivot and making the pivot equal to one.

Lesson Description:

Solving Ax = 0, Pivot Variables, and Special Solutions -- Lecture 7. How to solve some situations of Ax = 0 and use elimination to get a matrix into reduced row echelon form.

Gilbert Strang, 18.06 Linear Algebra, Spring 2005. (Massachusetts Institute of Technology: MIT OpenCourseWare), http://ocw.mit.edu (Accessed November 16, 2008). License: Creative Commons BY-NC-SA.
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