MATH 7101 – Advanced Linear Algebra Fall 2016

Instructor Amer Iqbal Email iqbal.amer@gmail.com; amer@alum.mit.edu

Course Basics Credit Hours 3 Lectures Number of Lectures in a Week 2 Duration 90 min Tutorials Number of Tutorials in a Week

(after first 4 lectures) 0 Duration

COURSE DESCRIPTION This course builds on the undergraduate linear algebra course. The topics covered will be: Vector Spaces, Subspaces, Direct Sums, Relation with Matrices, Linear Transformations, Kernel and Image of a Linear Transformation, TheRank PlusNullity Theorem, Linear transformation and matrices, Invariant Subspaces, Quotient Spaces, Isomorphism Theorems, Dual Basis, Operator Adjoints, Eigenvalues and eigenvectors, Characteristic polynomial, Diagonal form, Jordan canonical form, BilinearMaps, Tensor Products, Linear Transformations on a Tensor Product, Tensor Product of Linear Transformations, The Symmetric andAntisymmetric tensor products. Applications to various problems of above mentioned topics will be provided as well.

COURSE PREREQUISITE(S)

Undergraduate linear algebra course

COURSE OBJECTIVES This course is about linear vector space and various structures defined on them. It should be thought of as a continuation of the undergraduate linear algebra course but with more emphasis on abstract reasoning rather than matrix calculations. This course will explain to students the relation between vector spaces and matrices, the use of linear transformations to study various properties of the underlying vector spaces, various isomorphism theorems associated with linear transformations and how new vectors spaces can be constructed using the idea of tensor products.

Learning Outcomes At the end of the course students should be able to understand the basic features of a vector space and linear transformations defined on them and understand the isomorphism theorems. They should be able to understand the decomposition of the vector space into various invariant subspaces and the relationship of this decomposition with the Jordan canonical form. They should also understand the tensor product of vectors spaces and determine its basis and the corresponding matrices. Home Work: 10% Class Presentation: 20% Mid­Term: 20% Final Exam: 50%

Examination Detail

Midterm Exam

A comprehensive examination of 2 hours

Final Exam

A comprehensive examination of 3 hours

      COURSE OVERVIEW Week Topics Particulars

1­2 Review of Matrix Algebra and Vector Spaces Vector Spaces, Basis, Matrices, Basis Change, Linear Transformations, Matrix of LT, Dual Spaces, Basis, Subspaces, Direct Sums, Spanning Sets and Linear Independence, The Dimension of a Vector Space, The Complexification of a Real Vector Space

2­5 Linear Transformations Linear Transformations, Kernel and Image of LT, Isomorphisms, The Rank Plus Nullity Theorem, Change of Bases for Linear Transformations, Invariant Subspaces and Reducing Pairs

6­8 Isomorphism Theorems (Chapter 3 of the textbook) Quotient Spaces, Universal Property of Quotients, First Isomorphism Theorem, Complements, Linear Functionals, Dual Bases, Operator Adjoints

7­8 Eigenvalues & Eigenvectors The Characteristic Polynomial, The Rational Canonical Form, Geometric and Algebraic Multiplicities, Jordan Canonical Form, Galois group of a polynomial.

9­10 Inner Product Spaces Norm, Isometries, Orthogonality, Orthogonal and Orthonormal Sets, The Projection Theorem, The Riesz Representation Theorem

11­12 Bilinear Forms Symmetric, Skew­Symmetric and Alternate Forms, The Matrix of a Bilinear Form, Orthogonal Projections, Linear Functionals, Orthogonal Complements and Orthogonal Direct Sums, Isometries, Hyperbolic Spaces

13­14 Tensors Bilinear Maps,Tensor Products, Defining Linear Transformations on a Tensor Product, The Tensor Product of Linear Transformations, Multilinear Maps and Iterated Tensor Products

Textbooks/Supplementary Reading

1. Steven Roman, “Advanced Linear Algebra”, Published by Springer (2008). 2. Jonathan Golan, “The Linear Algebra a Beginning Graduate Student Ought to Know,” Published by Springer (2007).