18.06SC Linear Algebra (Fall 2011, MIT OCW): Lecture 08 - Solving Ax = 0: Pivot Variables, Special Solutions

18.06SC Linear Algebra

18.06SC Linear Algebra (Fall 2011, MIT OCW). Taught by Prof. Gilbert Strang, this course covers matrix theory and linear algebra, emphasizing topics useful in other disciplines such as physics, economics and social sciences, natural sciences, and engineering. This course has been designed for independent study. It provides everything you will need to understand the concepts covered in the course. The materials include: a complete set of lecture videos, summary notes, problem solving videos, and a full set of exams and solutions. (from ocw.mit.edu)

Lecture 08 - Solving Ax = 0: Pivot Variables, Special Solutions

We apply the method of elimination to all matrices, invertible or not. Counting the pivots gives us the rank of the matrix. Further simplifying the matrix puts it in reduced row echelon form R and improves our description of the null space.

References

Solving Ax = 0: Pivot Variables, Special Solutions
We apply the method of elimination to all matrices, invertible or not. Lecture Video and Summary. Suggested Reading. Problem Solving Video.

Go to the Course Home or watch other lectures:

I. Ax=b and the Four Subspaces

Lecture 01 - The Geometry of Linear Equations

Lecture 02 - An Overview of Key Ideas

Lecture 03 - Elimination with Matrices

Lecture 04 - Multiplication and Inverse Matrices

Lecture 05 - Factorization into A = LU

Lecture 06 - Transposes, Permutations, Vector Spaces

Lecture 07 - Column Space and Nullspace

Lecture 08 - Solving Ax = 0: Pivot Variables, Special Solutions

Lecture 09 - Solving Ax = b: Row Reduced form R

Lecture 10 - Independence, Basis and Dimension