Learn · Linear Algebra Through Geometry
Lecture 18 of 62Week 4 Lines in 2D, Parametric equations of line, Linear Maps in 2D, Linear Systems (2x2), eigenvalues and…
Determinant of 2x2 matrix as Area of Parallelogram, Determinant of linear transformations
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)
- Licence
- CC BY-NC-SA, as NPTEL states it. Non-commercial use only.
- Attribution
- NPTEL states: "Distributed under Creative Commons Attribution-ShareAlike / CC BY - NC - SA", beside a BY-NC-SA badge.
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- Tweens profiles and older. How courses are placed
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