MATH 453--LINEAR ALGEBRA II--SPRING 2020--A. FREIRE

Syllabus

1/9   Vector spaces, subspaces, linear transformations: definitions. (1B, 1C, 3A)
Examples: nullspace and range(3B), subspace spanned by a finite set, row and column space of a matrix
Column space=range, row space=orthogonal complement of the nullspace
HW 1: 1.C.12, 3.A.8, 3.A.9 (due 1/16)

1/14 linear independence, basis, existence of bases: cardinality of an LI set is bounded by the cardinality of a spanning set
(2A,2B,2C) HW 1: 2.A.15 (due 1/16)
HW1 solutions

1/16  isomorphisms and invertibility (3D)/ sum and direct sum of subspaces (1C,2C), product of v. spaces (3E)
rank+nullity theorem

1/21 applications: row rank=col rank (dimension of orth. complement), subspace dimension formula, dimension of space of linear maps
HW2 problems (due 1/23): 2A.3, 2C. 12, 3B.1, 3B.5

1/23: review problems from ch 1--3
HW3 (due 1/30): 1C.21, 2A.16, 3B.12, 3B.15, 3D.4

Review problems (boldface: HW or seen in class)
1C: 20,21,23
2A. 16
2C: 3, 8, 9, 10, 11,12, 13
3A: 11, 12, 13, 14
3B: 2, 4 to 8, 12, 13, 14, 15, 16
3D: 2, 4, 5, 6, 10, 16, 18
3E: 8, 9

CHAPTER 4 (Polynomials): read by 1/28. Review problems 2, 3, 4, 5, 7 (self-study)
C. Fefferman's proof of the Fundamental Theorem of Algebra
(Am. Math Monthly 1967)-- an elementary proof that doesn't use Complex Analysis

1/28: invariant subspaces, eigenvalues, triangular matrices (Ch.5)

1/30: complex triangular form, diagonalizability (Ch.5)

2/4: properties of triangular matrices, proof of the FTA, real triangular form (start: dimension two)
HW 4 problems: 5A.9, 5A.14, 5B.4, 5B.20
HW 4 solutions
practice problems: 5A: 10, 12, 13, 18, 20, 22, 23, 24, 25, 26, 27, 28, 29/ 5B: 3, 8, 9, 11, 12, 18/5C: 5, 6, 8, 12, 13, 15

2/6: QUIZ (1 HW problem from 2/3 list, 1 question from Ch.5), real triangular form (dimension 3 and higher), practice problems (Ch. 5)

2/11: problems from Chapter 5 (see above list)

2/13: geometry of linear maps of the plane/ inner product spaces/normed vector spaces/examples, Cauchy-Schwartz, triangle inequality
HW 5 (due 2/20): 6A-- 2, 5, 9, 15, 16

2/18: orthonormal bases, parallelogram law, orthogonal complement, orthogonal projection
From norms to inner products via the parallelogram law
(Guided sequence of four exercises, proving that a norm satisfying the Parallelogram Law comes from an inner product.)
--Extra-credit homework, due Tuesday 2/25.

2/20 minimization problems, adjoint of a linear map (7A)
HW 6: 6B: 14, 6C: 8, 12 7A: 8,9

2/25self-adjoint and normal operators (7A), spectral theorem (7B)

2/27 take-home midterm given (due Tuesday 3/3)
change of basis formula (10.7 on p.298), normal operators on real vector spaces (block-spectral theorem, 9B)
take-home midterm
midterm solutions

3/3 positive operators, isometries (7C)
HW 7 (due 3/5): (7C) 2, 7, 11, 12

3/5 polar decomposition, singular value decompositiom (7D)
Singular value decomposition for linear maps
(loosely based on Strang's book.)

3/10 review problems from Ch. 7
Problems: 7A 1,7,12,16, 19 7B: 2,6,9 7C: 6, 9 7D: 2,3,4,5,7,10,15,16,18
HW 8 (due 3/12): (7A) 5, 7, 19/ (7D) 4,5

3/12 QUIZ 2--one of the problems from the 3/10 list, plus a question.
Quadratic forms and minimizing properties of eigenvalues
Latex source code
(this may help those learning or recalling basic Latex)
(includes 5 problems=HW9, due 3/26)
Alternative: 3 problems from handout and porblems 2,10 from 8A
HW9 solutions

3/17, 3/19: SPRING BREAK

3/24 Nilpotent operators, generalized eigenspaces (Zoom) (8A)
Lecture notes (scan, 4 pages)
(8A) review problems: 5, 6, 9, 10, 11, 12, 13, 14, 15, 16 to 20.

3/26 Basis of generalized eigenvectors/matrix interp./square roots (8B) (Zoom)
Lecture (scan, 7 pages)
HW 10 (due 4/2): (8B) 1, 2, 3, 6, 7

3/31 Jordan form, complex case (8D)
Lecture (scan, 7 pages)

4/2 Jordan form, minimal polynomial (8C)
Lecture (scan, 6 pages)

4/7 Characteristic polynomial (8C)/ problems
Review problems: (8C), 1 to 16.
Lecture (scan, 5 pages)

4/9 no class day (University-wide): optional midterm (online)
Topics: Chapters 7 and 8, handout "quadratic forms"
optional midterm       solutions

4/14 complexification/ (9A) Real Jordan form
Lecture (scan, 8 pages)      page 7
Review problems (9A): 3, 4, 9, 10, 11, 12, 13, 14, 15, 16

4/16 Real Jordan form: examples/matrix exponential
Lecture (scan. 7 pages)
HW set 11 (due 4/23)     solutions

4/21 matrix exponential: application/ determinant and trace
Lecture (scan, 8 pages)

4/23 Review problems
Lecture (scan, 5 pages)

FINAL EXAM: Tuesday, 4/28, 8:00-10:00 AM (comprehensive, proctored-timed Zoom test.)
Final Exam    solutions