Applied Linear Algebra

Week 1:
Lecture 1: Why should we care about algebra?
Lecture 2: Uses of linear algebra in different domains
Lecture 3: Power of abstraction and geometric insights
Lecture 4: Equivalent systems of linear equations
Lecture 5: Row reduced form
Lecture 6: Row reduced echelon form.

Week 2:
Lecture 7: Solving for Ax=0
Lecture 8: Row rank of matrices
Lecture 9: Groups and Abelian Groups
Lecture 10: Rings, integral domains, and fields
Lecture 11: Fields: Examples and properties
Lecture 12: Vector Spaces

Week 3:
Lecture 13: Examples of vector spaces
Lecture 14: Subspaces
Lecture 15: Examples of subspaces
Lecture 16: Sum and intersection of subspaces
Lecture 17: Span and linear independence
Lecture 18: Generating set and basis

Week 4:
Lecture 19: Properties of basis
Lecture 20: Dimension of a vector space
Lecture 21: Dimensions of special subspaces and properties
Lecture 22: Co-ordinates and ordered basis
Lecture 23: Row and column rank
Lecture 24: Rank and nullity of matrices

Week 5:
Lecture 25: Linear transformations and operators
Lecture 26: Rank nullity theorem for linear transformations
Lecture 27: Injective, surjective and bijective linear mappings
Lecture 28: Isomorphism and their compositions
Lecture 29: Linear transformations under change of basis
Lecture 30: Linear functionals

Week 6:
Lecture 31: Dual basis and dual maps
Lecture 32: Annihilators, double duals
Lecture 33: Products of vector spaces
Lecture 34: Quotient spaces
Lecture 35: Quotient maps
Lecture 36: First isomorphism theorem

Week 7:
Lecture 37: Inner product spaces
Lecture 38: Examples of inner products
Lecture 39: Cauchy Schwarz and triangle inequalities
Lecture 40: Some results and applications of inner products (in solving Ax=b)
Lecture 41: Gram-Schmidt orthonormalization
Lecture 42: Best approximation of a vector in a subspace

Week 8:
Lecture 43: Orthogonal complements of subspaces and their properties
Lecture 44: Orthogonal projection map and its properties
Lecture 45: “Best” solution for Ax=b
Lecture 46: Applications of “best” solution
Lecture 47: Adjoint operators on inner product spaces
Lecture 48: Miscellaneous results on inner products and inner product spaces, and their applications (e.g. Haar wavelets, Fourier series)

Week 9:
Lecture 49: Solutions of linear second order differential equations and phase portraits
Lecture 50: Eigenvalues and eigen vectors
Lecture 51: Diagonalizability for self-adjoint operators
Lecture 52: Linear independence of eigen vectors and diagonalizability, evaluation of matrix functions
Lecture 53: Algebraic and geometric multiplicities
Lecture 54: Decomposition of a vector space into sums and direct sums of suitable subspaces

Week 10:
Lecture 55: Equivalent conditions for diagonalizability
Lecture 56: A-invariant subspaces: definition and examples
Lecture 57: Polynomials and their ideals
Lecture 58: Minimal polynomial
Lecture 59: Minimal polynomial and characteristic polynomial
Lecture 60: Further properties of minimal polynomial

Week 11:
Lecture 61: Bezout’s identity for polynomials
Lecture 62: Application of Bezout’s identity to coprime factors of minimal polynomial
Lecture 63: Recipe for best representation of non-diagonalizable linear operators
Lecture 64: Jordan canonical form
Lecture 65: Proof for Jordan canonical form
Lecture 66: Proof of Cayley Hamilton theorem

Week 12:
Lecture 67: Application of linear algebra to algebraic graph theory
Lecture 68: Properties of graph Laplacian matrix: Fiedler eigenvalue
Lecture 69: Consensus problem
Lecture 70: Solution of the agreement protocol
Lecture 71: Applications to opinion dynamics
Lecture 72: Further applications of linear algebra to multi-agent systems