Learn · Linear Algebra Through Geometry
Lecture 51 of 62Week 11 Real Symmetric Matrices, Quadratic forms
Eigen subspace and dimension
Prof. Ashok Rao, Prof. M Krishna Kumar and Prof. Arulalan M R · IISc Bangalore
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- Linear Algebra Through Geometry, IISc Bangalore (at NPTEL)
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- CC BY-NC-SA, as NPTEL states it. Non-commercial use only.
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- NPTEL states: "Distributed under Creative Commons Attribution-ShareAlike / CC BY - NC - SA", beside a BY-NC-SA badge.
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All the lectures
- Introduction to Linear Algebra and Matrices
- Geometry of System of linear equations - Straight lines and planes, Matrix Definitions
- Some Interpretations to solutions of system of linear equations
- Matrix Operations, Homogeneous system of equations
- Matrix Operations, Homogeneous system of equations
- Elementary Row Operations
- Elementary Row operations - How do they work?
- Determinant and Inverse of a matrix
- Interpreting the inverse of a matrix
- Cramer’s rule
- Points and Vectors in 2D
- Vector Length and properties
- Combining Vectors
- Linearly Independent and Dependent vectors, , Dot Product of vectors
- Angle between two vectors, Orthogonal projections
- Lines and Parametric Equations of lines, Linear Maps
- Rotation, Shear and Projection transformations
- Determinant of 2x2 matrix as Area of Parallelogram, Determinant of linear transformations
- System of 2 linear equations in 2 unknowns from vector perspective
- Eigenvalues and eigenvectors
- Vectors in 3D, Linear combination of vectors in 3D
- Projection of vector on another vector, onto a line passing through the origin, onto a plane passing through the origin
- Area of a parallelogram in 3D, Cross product
- Interpreting the cross-product, Properties of cross-product
- Volume of a parallelepiped, Lines in 3D, Intersection of line and plane
- Linear Maps in 3D - Scaling and Reflection
- Linear Maps in 3D - Reflection about a plane, Shear
- Rotation in 3D
- Determinant and its properties
- Eigenvalues and eigenvectors in 3D
- Linear systems in 3D and geometric perspective
- Homogeneous system in 3D
- LU Decomposition
- Least Squares Solution, Gram-Schmidt Orthogonalization, QR Decomposition
- Orthogonal Matrix, Linear Independence, eigenvalues and eigenvectors in 3D
- Vector Space and Properties
- Examples of vector spaces - Polynomial space, planes and lines through origin
- Vector Subspaces and their geometry
- Combining vectors in a vector space, Linear Independence
- Span, Basis, Dimension of a vector space, Fourier Expansion
- Homogeneous system of linear equations and null space of a matrix
- Column Space of A
- Subspaces associated matrix A transpose, Nullity, Rank
- Orthogonal Complement of a subspace
- Orientation of the four fundamental subspaces of a matrix A
- System of linear equations with no solution - Inconsistent systems
- Least squares solution, Pseudoinverse of A
- Projection and Projection Matrices
- Pseudoinverse of special matrices
- Eigen decomposition
- Eigen subspace and dimension
- Real Symmetric matrix and properties
- Eigenvalues and eigenvectors of real symmetric matrices
- Effect of a real symmetric matrix - Geometric Interpretation
- Spectral Theorem, Quadratic Forms
- Singular Value Decomposition
- Relationship between SVD and Eigen Decomposition
- An Interpretation of SVD
- Fourier Series & Transform through Linear Algebra
- Practical Applications of Linear Algebra 1
- Practical Applications of Linear Algebra 2
- Summary & Credits