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Applied Linear Algebra
Online
12 Weeks
Free Quick Facts
| Medium Of Instructions | Mode Of Learning | Mode Of Delivery |
|---|---|---|
| English | Self Study | Video and Text Based |
Courses and Certificate Fees
| Fees Informations | Certificate Availability | Certificate Providing Authority |
|---|---|---|
| INR 1000 | yes | IIT Bombay |
The Syllabus
- Why should we care about algebra?
- Uses of linear algebra in different domains
- Power of abstraction and geometric insights
- Equivalent systems of linear equations
- Row reduced form
- Row reduced echelon form
- Solving for Ax=0
- Row rank of matrices
- Groups and Abelian Groups
- Rings, integral domains, and fields
- Fields: Examples and properties
- Vector Spaces
- Examples of vector spaces
- Subspaces
- Examples of subspaces
- Sum and intersection of subspaces
- Span and linear independence
- Generating set and basis
- Properties of basis
- Dimension of a vector space
- Dimensions of special subspaces and properties
- Co-ordinates and ordered basis
- Row and column rank
- Rank and nullity of matrices
- Linear transformations and operators
- Rank nullity theorem for linear transformations
- Injective, surjective and bijective linear mappings
- Isomorphism and their compositions
- Linear transformations under change of basis
- Linear functionals
- Dual basis and dual maps
- Annihilators, double duals
- Products of vector spaces
- Quotient spaces
- Quotient maps
- First isomorphism theorem
- Inner product spaces
- Examples of inner products
- Cauchy Schwarz and triangle inequalities
- Some results and applications of inner products (in solving Ax=b)
- Gram-Schmidt orthonormalization
- Best approximation of a vector in a subspace
- Orthogonal complements of subspaces and their properties
- Orthogonal projection map and its properties
- “Best” solution for Ax=b
- Applications of “best” solution
- Adjoint operators on inner product spaces
- Miscellaneous results on inner products and inner product spaces, and their applications (e.g. Haar wavelets, Fourier series)
- Solutions of linear second order differential equations and phase portraits
- Eigenvalues and eigen vectors
- Diagonalizability for self-adjoint operators
- Linear independence of eigen vectors and diagonalizability, evaluation of matrix functions
- Algebraic and geometric multiplicities
- Decomposition of a vector space into sums and direct sums of suitable subspaces
- Equivalent conditions for diagonalizability
- A-invariant subspaces: definition and examples
- Polynomials and their ideals
- Minimal polynomial
- Minimal polynomial and characteristic polynomial
- Further properties of minimal polynomial
- Bezout’s identity for polynomials
- Application of Bezout’s identity to coprime factors of minimal polynomial
- Recipe for best representation of non-diagonalizable linear operators
- Jordan canonical form
- Proof for Jordan canonical form
- Proof of Cayley Hamilton theorem
- Application of linear algebra to algebraic graph theory
- Properties of graph Laplacian matrix: Fiedler eigenvalue
- Consensus problem
- Solution of the agreement protocol
- Applications to opinion dynamics
- Further applications of linear algebra to multi-agent systems
Instructors
Articles
31 Aug'25, 03:44 PM
17 Jun'25, 10:59 AM
28 Feb'24, 08:51 AM
28 Feb'24, 07:16 AM
31 Aug'25, 03:44 PM
17 Jun'25, 10:59 AM
28 Feb'24, 08:51 AM
28 Feb'24, 07:16 AM