# InfoCoBuild

## 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 11 - The Four Fundamental Subspaces

For some vectors b the equation Ax = b has solutions and for others it does not. Some vectors x are solutions to the equation Ax = 0 and some are not. To understand these equations we study the column space, nullspace, row space and left nullspace of the matrix A.

 References The Four Fundamental Subspaces For some vectors b the equation Ax = b has solutions and for others it does 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 Lecture 11 - The Four Fundamental Subspaces Lecture 12 - Matrix Spaces; Rank 1; Small World Graphs Lecture 13 - Graphs, Networks, Incidence Matrices Lecture 14 - Exam 1 Review II. Least Squares, Determinants and Eigenvalues Lecture 15 - Orthogonal Vectors and Subspaces Lecture 16 - Projections onto Subspaces Lecture 17 - Projection Matrices and Least Squares Lecture 18 - Orthogonal Matrices and Gram-Schmidt Lecture 19 - Properties of Determinants Lecture 20 - Determinant Formulas and Cofactors Lecture 21 - Cramer's Rule, Inverse Matrix and Volume Lecture 22 - Eigenvalues and Eigenvectors Lecture 23 - Diagonalization and Powers of A Lecture 24 - Differential Equations and exp(At) Lecture 25 - Markov Matrices; Fourier series Lecture 26 - Exam 2 Review III. Positive Definite Matrices and Applications Lecture 27 - Symmetric Matrices and Positive Definiteness Lecture 28 - Complex Matrices; Fast Fourier Transform Lecture 29 - Positive Definite Matrices and Minima Lecture 30 - Similar Matrices and Jordan Form Lecture 31 - Singular Value Decomposition Lecture 32 - Linear Transformations and their Matrices Lecture 33 - Change of Basis; Image Compression Lecture 34 - Left and Right Inverses; Pseudoinverse Lecture 35 - Exam 3 Review Lecture 36 - Final Course Review