WELCOME TO MAT 315   Advanced LinearAlgebra

Spring 2022

Time and place:

Lecture: TuTh 3:00-4:20PM in Light Engineering 152

Recitation: W. 11:45AM-12:40PM in Light Engineering 152

Introduction: This is an advanced mathematically rigorous course with complete proofs.  Topics covered will include vector spaces, their direct sums and tensor products,  linear transformations, eigenvalues, dual spaces and inner products, bilinear and quadratic functions and quotient vector spaces  Also the trace, determinant, charateristic and minimal polynomial of a linear operator and the Jordan form if time permits.

Text Book: Linear Algebra Done Right   ( 3nd edition) by Sheldon Axler, Springer,  (c) 2015

 Instructor: Prof. David Ebin Math Tower 5-107 tel. 632-8283 E-mail: ebin@math.sunysb.edu Office Hours: Tuesday or Thursday 1:30-2:50,  or by appointment

Assistant: Daniil Glukhovskiy

Homework: Homework will be assigned every week.  Doing the homework is a fundamental part of the course work.  Problems should be handed in in your recitation section.

Assignment due Feb. 23: page 57 problems 2, 3, 4, 11, 12; page 67 problems 1, 2, 4, 5, 6

Assignment due  March 2: page 78, problems3, 6, 12 and 14

Assignment due March 9: page 88, problems 5, 19; page 99, problems 7, 12; page 113, problems 6, 8, 16, 27, 31

Assignment due March 23: page 113, problems 34 and 35; page129, problems 8, 9, 11

Grading Policy: The overall numerical grade will be computed by the formula   20% Homework + 15% first Midterm Exam + Second Midterm 25% +  40% Final Exam

First Midterm Exam: February 8 in class with MAT310

Topics for 2nd midterm exam: The vector space of linear maps, the rank plus nullity theorem also called the fundamental theorem of linear maps, the matrix of a linear map with respect to bases of the domain and of the range, how the matrix changes when the bases change,  proof that the matrix of a composition of linear maps is the product of the matrices of the maps,  products and quotients of vector spaces, dual spaces and duals of linear maps, the division algorithm for polynomials, fatorization of polynomials over the real and complex numbers, eigenvalues and eigenvectors, the space of linear operators of a vector space, change of basis and its effect on the matrix of an operator, similar matrices, polynomials of operators,  upper triangular matrices, eigenvalues and diagonal matrices

Second Midterm exam: Tuesday, March 29

Assignment due April 6: page 138, problems 1, 2, 7, 8, 11, 16, 23; page 153, problems 2, 4, 8; page 160, problems 1, 12 due April 6

Assignment due April 13: page 161 problem 16; page 175 problems 4, 5, 9, 12, 15, 19, 26ab, 28, 31 due April 13

Assignment due April 20: page 189, problems 2, 6, 7, 8, 13; page 201, problems 3, 5, 7, 10, 12 due April 20

Assignment due April 27: page 214, problems 1, 2, 5, 7, 14; page 223, problems 1, 2, 6, 11, 12 due April 27

Last Assignment due May 4: page 231, problems 2, 9, 11, 13; page 249, problems 3, 5, 12 due May 4

Final Exam: Tuesday, May 10, 2:15-5:00PM

Topics for final exam: All topics for the second midterm, Bases of a vector space and dual bases of the dual space, proof that if M is the matrix of a linear map with respect to bases of its domain and range, then the matrix of the dual map with respect to dual bases is the transpose of M.  Column and row rank of a matrix, proof that they are equal, Complex numbers and their absolute value, complex conjugate, Triangle inequality for complex numbers, vector spaces over finite fields, polynomials with coefficients in finite fields, Quotient spaces, Inner product spaces and norms of vectors and metrics on vector spaces,  Schwartz inequality and its proof, Orthogonal vectors, orthogonal and orthonormal bases, Parallelogram equality, Gram-Schmidt procedure to get orthonormal sets of vectors,  particularly polynomials,   Linear operators, If V is a complex vector space and T is a linear operator on V, Find a basis of V that makes the matrix of T upper triangular, Find such an orthonormal basis, The representation theorem for vector spaces with an inner product, Subspaces and orthogonal subspaces,  Normal and self-adjoint operators,  Hermitian matrices, Proof that for complex vector spaces, normal operators have an orthonormal basis of eigenvectors,  Same for self-adjoint operators in real vector spaces, positive operators and positive square roots of operators, Isometries and their properties, null spaces of powers of an operator, Eigenspaces and generalized eigenspaces, nilpotent operators, Proof that every nilpotent operator operator has a strictly upper-triangular matrix with respect to some basis,  If  T: V --> V  is a linear operator (V complex) with eigenvalues a_1, a_2, ..., a_k, prove that V is a direct sum of the generalized eigenspaces,  G(a_1, T), G(a_2, T), ..., G(a_k, T).  Define the characteristic and minimal polynomials of T,  Prove the Hamilton-Jacobi theorem, and prove that the minimal polynomial divides the characteristic polynomial,  Define a Jordan basis for an operator T: V --> V.

N. B. Use of calculators is not permitted in any of the examinations.

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