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6. Column Space and Nullspace
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Classroom Contents
Linear Algebra
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- 1 An Interview with Gilbert Strang on Teaching Linear Algebra
- 2 Course Introduction | MIT 18.06SC Linear Algebra
- 3 1. The Geometry of Linear Equations
- 4 Geometry of Linear Algebra
- 5 Rec 1 | MIT 18.085 Computational Science and Engineering I, Fall 2008
- 6 An Overview of Key Ideas
- 7 2. Elimination with Matrices.
- 8 Elimination with Matrices
- 9 3. Multiplication and Inverse Matrices
- 10 Inverse Matrices
- 11 4. Factorization into A = LU
- 12 LU Decomposition
- 13 5. Transposes, Permutations, Spaces R^n
- 14 Subspaces of Three Dimensional Space
- 15 6. Column Space and Nullspace
- 16 Vector Subspaces
- 17 7. Solving Ax = 0: Pivot Variables, Special Solutions
- 18 Solving Ax=0
- 19 8. Solving Ax = b: Row Reduced Form R
- 20 Solving Ax=b
- 21 9. Independence, Basis, and Dimension
- 22 Basis and Dimension
- 23 10. The Four Fundamental Subspaces
- 24 Computing the Four Fundamental Subspaces
- 25 11. Matrix Spaces; Rank 1; Small World Graphs
- 26 Matrix Spaces
- 27 12. Graphs, Networks, Incidence Matrices
- 28 Graphs and Networks
- 29 13. Quiz 1 Review
- 30 Exam #1 Problem Solving
- 31 14. Orthogonal Vectors and Subspaces
- 32 Orthogonal Vectors and Subspaces
- 33 15. Projections onto Subspaces
- 34 Projection into Subspaces
- 35 16. Projection Matrices and Least Squares
- 36 Least Squares Approximation
- 37 17. Orthogonal Matrices and Gram-Schmidt
- 38 Gram-Schmidt Orthogonalization
- 39 18. Properties of Determinants
- 40 Properties of Determinants
- 41 19. Determinant Formulas and Cofactors
- 42 Determinants
- 43 20. Cramer's Rule, Inverse Matrix, and Volume
- 44 Determinants and Volume
- 45 21. Eigenvalues and Eigenvectors
- 46 Eigenvalues and Eigenvectors
- 47 22. Diagonalization and Powers of A
- 48 Powers of a Matrix
- 49 23. Differential Equations and exp(At)
- 50 Differential Equations and exp (At)
- 51 24. Markov Matrices; Fourier Series
- 52 Markov Matrices
- 53 24b. Quiz 2 Review
- 54 Exam #2 Problem Solving
- 55 25. Symmetric Matrices and Positive Definiteness
- 56 Symmetric Matrices and Positive Definiteness
- 57 26. Complex Matrices; Fast Fourier Transform
- 58 Complex Matrices
- 59 27. Positive Definite Matrices and Minima
- 60 Positive Definite Matrices and Minima
- 61 28. Similar Matrices and Jordan Form
- 62 Similar Matrices
- 63 29. Singular Value Decomposition
- 64 Computing the Singular Value Decomposition
- 65 30. Linear Transformations and Their Matrices
- 66 Linear Transformations
- 67 31. Change of Basis; Image Compression
- 68 Change of Basis
- 69 33. Left and Right Inverses; Pseudoinverse
- 70 Pseudoinverses
- 71 32. Quiz 3 Review
- 72 Exam #3 Problem Solving
- 73 34. Final Course Review
- 74 Final Exam Problem Solving
