WELCOME TO MAT 315

Time and place:
Lecture: TuTh 3:004:20PM in Light Engineering 152
Recitation: W. 11:45AM12:40PM 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 5107 tel. 6328283 Email: ebin@math.sunysb.edu Office Hours: Tuesday or Thursday 1:302:50, or by appointment 
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 29Assignment 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, 2:155:00PM
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,
GramSchmidt 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 selfadjoint operators, Hermitian
matrices, Proof that for complex vector spaces, normal operators have an
orthonormal basis of eigenvectors, Same for selfadjoint 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
uppertriangular 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 HamiltonJacobi theorem, and prove that the minimal
polynomial divides the characteristic polynomial, Define a Jordan
basis for an operator T: V > V.
Disabilities: If you have a physical, psychological, medical, or learning disability that may impact your course work, please contact Disability Support > Services or call (631) 6326748. They will determine with you what accommodations are necessary and appropriate. All information and documentation is confidential. Students who require assistance during emergency evacuation are encouraged to discuss their needs with their professors and Disability Support Services. For procedures and information go to the Evacuation Guide for People with Physical Disabilities.
Academic Integrity: Each student must pursue his or her academic goals honestly and be personally accountable for all submitted work. Representing another person's work as your own is always wrong. Faculty are required to report any suspected instances of academic dishonesty to the Academic Judiciary. For more comprehensive information on academic integrity, including categories of academic dishonesty, please refer to the academic judiciary website.
Critical Incident management: Stony Brook University expects students to respect the rights, privileges, and property of other people. Faculty are required to report to the Office of Judicial Affairs any disruptive behavior that interrupts their ability to teach, compromises the safety of the learning environment, or inhibits students' ability to learn.