Lecture 40 of 158Orthogonal Matrices and Gram-Schmidt10 min
Problem Solving: Gram-Schmidt Orthogonalization
Prof. Gilbert Strang · Massachusetts Institute of Technology
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- Linear Algebra, Massachusetts Institute of Technology (at MIT OpenCourseWare)
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- CC BY-NC-SA 4.0, as MIT OpenCourseWare states it. Non-commercial use only.
- Attribution
- Prof. Gilbert Strang. Linear Algebra. Fall 2011. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu/. License: Creative Commons BY-NC-SA.
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All the lectures
- Linear Algebra Course Introduction
- An Interview with Gilbert Strang on Teaching Linear Algebra
- The Geometry of Linear Equations
- Geometry of Linear Algebra
- 线性代数的几何表示
- An Overview of Linear Algebra
- An Overview of Key Ideas
- Elimination with Matrices
- Recitation: Elimination with Matrices
- Multiplication and Inverse Matrices
- Inverse Matrices
- Factorization into A = LU
- LU Decomposition
- Transposes, Permutations, Vector Spaces
- Subspaces of Three Dimensional Space
- 三维空间的子空间
- Column Space and Nullspace
- Vector Subspaces
- Solving Ax = 0: Pivot Variables, Special Solutions
- Solving Ax=0
- Solving Ax = b: Row Reduced Form R
- Solving Ax=b
- Independence, Basis and Dimension
- Basis and Dimension
- The Four Fundamental Subspaces
- Computing the Four Fundamental Subspaces
- Matrix Spaces; Rank 1; Small World Graphs
- Matrix Spaces
- Graphs, Networks, Incidence Matrices
- Graphs and Networks
- Exam 1 Review
- Exam 1 Problem Solving
- Orthogonal Vectors and Subspaces
- Problem Solving: Orthogonal Vectors and Subspaces
- Lecture 15: Projections onto Subspaces
- Problem Solving: Projection onto Subspaces
- Lecture 16: Projection Matrices and Least Squares
- Problem Solving: Least Squares Approximation
- Lecture 17: Orthogonal Matrices and Gram-Schmidt
- Problem Solving: Gram-Schmidt Orthogonalization
- Lecture 18: Properties of Determinants
- Problem Solving: Properties of Determinants
- Lecture 19: Determinant Formulas and Cofactors
- Problem Solving: Determinants
- 行列式
- Lecture 20: Cramer's Rule, Inverse Matrix and Volume
- Problem Solving: Determinants and Volume
- 行列式和体积
- Lecture 21: Eigenvalues and Eigenvectors
- Problem Solving: Eigenvalues and Eigenvectors
- Lecture 22: Diagonalization and Powers of A
- Problem Solving: Powers of a Matrix
- Lecture 23: Differential Equations and exp(At)
- Problem Solving: Differential Equations and exp(At)
- 微分方程指数矩阵(At)
- Lecture 24: Markov Matrices; Fourier Series
- Problem Solving: Markov Matrices
- Exam 2 Review
- Exam 2 Problem Solving
- 第二次考试例题
- Lecture 25: Symmetric Matrices and Positive Definiteness
- Problem Solving: Symmetric Matrices and Positive Definiteness
- Lecture 26: Complex Matrices; Fast Fourier Transform (FFT)
- Problem Solving: Complex Matrices
- Lecture 27: Positive Definite Matrices and Minima
- Problem Solving: Positive Definite Matrices and Minima
- Lecture 28: Similar Matrices and Jordan Form
- Problem Solving: Similar Matrices
- Lecture 29: Singular Value Decomposition
- Problem Solving: Computing the Singular Value Decomposition
- Lecture 30: Linear Transformations and their Matrices
- Problem Solving: Linear Transformations
- Lecture 31: Change of Basis; Image Compression
- Problem Solving: Change of Basis
- Lecture 33: Left and Right Inverses; Pseudoinverse
- Problem Solving: Pseudoinverses
- Exam 3 Review
- Exam 3 Problem Solving
- Final Course Review
- Final Exam Problem Solving
- Gil Strang's Final 18.06 Linear Algebra Lecture
- The Geometry of Linear Equations
- Geometry of Linear Algebra
- 线性代数的几何表示
- An Overview of Linear Algebra
- An Overview of Key Ideas
- Lecture: Elimination with Matrices
- Elimination with Matrices
- Multiplication and Inverse Matrices
- Inverse Matrices
- Factorization into A = LU
- LU Decomposition
- Transposes, Permutations, Vector Spaces
- Subspaces of Three Dimensional Space
- 三维空间的子空间
- Column Space and Nullspace
- Vector Subspaces
- Solving Ax = 0: Pivot Variables, Special Solutions
- Solving Ax=0
- Solving Ax = b: Row Reduced Form R
- Solving Ax=b
- Independence, Basis and Dimension
- Basis and Dimension
- The Four Fundamental Subspaces
- Computing the Four Fundamental Subspaces
- Matrix Spaces; Rank 1; Small World Graphs
- Matrix Spaces
- Graphs, Networks, Incidence Matrices
- Graphs and Networks
- Exam 1 Review
- Exam 1 Problem Solving
- Orthogonal Vectors and Subspaces
- Problem Solving: Orthogonal Vectors and Subspaces
- Lecture 15: Projections onto Subspaces
- Lecture 16: Projection Matrices and Least Squares
- Problem Solving: Least Squares Approximation
- Lecture 17: Orthogonal Matrices and Gram-Schmidt
- Problem Solving: Gram-Schmidt Orthogonalization
- Lecture 18: Properties of Determinants
- Problem Solving: Properties of Determinants
- Lecture 19: Determinant Formulas and Cofactors
- Problem Solving: Determinants
- 行列式
- Lecture 20: Cramer's Rule, Inverse Matrix and Volume
- Problem Solving: Determinants and Volume
- 行列式和体积
- Lecture 21: Eigenvalues and Eigenvectors
- Problem Solving: Eigenvalues and Eigenvectors
- Lecture 22: Diagonalization and Powers of A
- Problem Solving: Powers of a Matrix
- Lecture 23: Differential Equations and exp(At)
- Problem Solving: Determinants and VolumeDifferential Equations and exp(At)
- 微分方程指数矩阵(At)
- Lecture 24: Markov Matrices; Fourier Series
- Problem Solving: Markov Matrices
- Exam 2 Review
- Exam 2 Problem Solving
- 第二次考试例题
- Lecture 25: Symmetric Matrices and Positive Definiteness
- Problem Solving: Symmetric Matrices and Positive Definiteness
- Lecture 26: Complex Matrices; Fast Fourier Transform (FFT)
- Problem Solving: Complex Matrices
- Lecture 27: Positive Definite Matrices and Minima
- Problem Solving: Positive Definite Matrices and Minima
- Lecture 28: Similar Matrices and Jordan Form
- Problem Solving: Similar Matrices
- Lecture 29: Singular Value Decomposition
- Problem Solving: Determinants and VolumeComputing the Singular Value Decomposition
- Lecture 30: Linear Transformations and their Matrices
- Problem Solving: Linear Transformations
- Lecture 31: Change of Basis; Image Compression
- Problem Solving: Change of Basis
- Lecture 33: Left and Right Inverses; Pseudoinverse
- Problem Solving: Pseudoinverses
- Exam 3 Review
- Exam 3 Problem Solving
- Final Course Review
- Final Exam Problem Solving