math54-f22:start

# Math 54 Honor (2022 Fall)

Course Number 24291

Lecture:
TuTh 11:00A-12:29P - 289 - Cory
Instructor: Peng Zhou, pzhou.math@gmail.com
Zoom: Meeting ID: 926 4346 7269, Passcode: 736211
Office Hours: Tue: 12:40 - 2:00pm. Wed: 4-5pm. Evan 753

Discussion Session:
MWF 11:00A-11:59A - 289 - Cory
Instructor: Sergio Escobar
Office Hours: Mon, Friday 12:10 - 1:00pm Evans 842

First lecture is on Aug 25 (Thu). First discussion session on Friday Aug 26 (Fri)

## Online Tools

We will have homework, quiz, participation, midterm and final.

• Homework will not be graded, you are encouraged to try them, discuss about them on our online discord forum.
• Quiz 20%. We will have weekly quizzes on Monday Wednesday discussion session.
• Participation 10%. It includes: interaction during class and discussion, asking and answering questions on our discord forum, sharing lecture notes or homework solutions, or any useful materials for learning.
• 2 Midterm 15% + 15%, Final 40%.

## Lectures

• Lecture 1 (Aug 25): Givental's book, section 1: Vector.
• HW 1: page 7 in [LA], pick at least 4 problems from 1-12 and solve it. (Givental's exercises are too interesting to be put in the quiz. So our quiz will be decoupled from the HW actually. Our first quiz will be a test run, and no grade will be recorded.)
• Discussion (Aug 29): quiz 1, solution
• Lecture 2 (Aug 30): section 2: analytic geometry: conic curves, linear transformations.
• Lecture 3 (Sep 1): review. affine space and affine linear transformation. Set theoretic notation. (ODE section 1.3)
• HW 2: [ODE] (conic curve:)p11, read the Example. (matrices:)Ex 1.3.1 (b,c,d,g). Ex 1.3.2 (a-f)
• Lecture 4 (Sep 6): [ODE 1.3]. 2×2 matrices: determinant, inverse. Orthogonal transformation. video
• Lecture 5 (Sep 8): [LA] Ch 1 Sec 3. Orthogonal Transformations. Complex Numbers. video
• HW 3:
• [ODE]1.3.4, (The notion of similar matrices are on top of page 19)
• [LA] p22, 44,45,47,48,49,51,54,57,58.
• quiz will be about complex numbers, similar to [LA] exercise above.
• Lecture 6 (Sep 13) [LA] 1.4: Four theorems in Linear Algebra (intro) video
• Lecture 7 (Sep 15) [LA] 2.1 Matrices, 2.2 Determinants audio only
• HW 4: : 86, 88, 91, 95, 98, 99, 100, 107, 111, 112, 114, 116, 117, 119
• Lecture 8 (Sep 20) [LA] 2.2 Determinants. 5 properties of det(A), $\det(AB)= \det(A) \det(B)$. video
• Lecture 9 (Sep 22) [LA] finish up 2.2 about cofactor and block matrix multiplication video
• HW 5: 110, 118, 132, 135, 138, 139, 140, 143
• Lecture 9 (Sep 27) [LA] 2.3 Abstract Vector spaces. Fields. video
• Lecture 10 (Sep 29) [LA] 3.1 Dimension and Ranks video
• HW 6: 150,151,152,155,159(notice $W^\perp$ in general lives in the dual space),160, 165, 167
• Lecture 11 (Oct 4) [LA] 3.1. rank theorem video
• Midterm 1: Oct 5. 11:10-12 , one page (both side) cheat-sheet allowed
• Lecture 12 (Oct 6) [LA] 3.1 (still rank theorem) video
• HW 7: [LA] 178, 179, 180, 184, 185, 187
• Give an exposition of the Remark in page 79 (the one after Corollary 1)
• 195 (this one is a bit hard, try a few examples first)
• Lecture 13 (Oct 11) [LA] 3.2 Gauss Elimination. Echelon form video
• Post Midterm 1 Feedback form(anonymous)
• Lecture 14 (Oct 13) [LA] Q&A on dual vector space, dual basis and quotient space. video
• HW 8:
• [LA] read about LPU decomposition. give some 2×2 and 3×3 examples where the permutation matrices is not the identity. read about flag manifold
• 197, 199, 201, 202, 206*(flag variety), 208*. *=extra
• Lecture 15 (Oct 18) [LA] 3.2 LPU decomposition. Flag varietyvideo
• Lecture 16 (Oct 20) [LA] 3.3 Quadratic form video
• HW 9: (1) [LA] 211 - 219. (2) Turn the proof of Lemma of existence of orthogonal basis, into an algorithm to find an orthogonal basis. Namely, given a symmetric matrix B of size $n \times n$, find a matrix $A$, such that $A^t B A$ is diagonal.
• Lecture 17 (Oct 25): Sesquilinear form and Hermitian form. video
• Lecture 18 (Oct 27): [LA] Sylvester theorem. Orthogonalization preserving flag. video
• HW 10: 226, 229,230, 232, 233, 234, *239
• Lecture 19 (Nov 1): [LA] Cauchy-Schwarz. Adjoint map. Normal Operator. video
• Lecture 20 (Nov 3): [LA] Proof of Spectral theorem for normal operator. Complexification of real vector spaces. video
• HW 10: 253-257, 260, 261, 263,265
• Lecture 21 (Nov 8) [ODE] 2.1.1 and 2.1.2 video
• No quiz on Nov 9 (wednesday)
• Lecture 22 (Nov 10) [ODE] 2.1.2 and [LA] 4.2 Jordan form video
• Midterm 2. Nov 14 Monday in discussion. more info here, midterm 2 changed to online
• Lecture 23 (Nov 15) Jordan form video, note
• Lecture 24 (Nov 17) Constant Coeff Equation. note , video
• HW: [ODE] (1) page 68, problem 2,4. (2) Page 115, ex 3.6.1, (3) page 119 ex 3.6.2
• Lecture 25 (Nov 22) Boundary Conditions. Inhomogeneous Equations. note, video
• Midterm 2 Discussion (Nov 22 11:10-12:30). note, video
• HW: [ODE] p128, Ex 3.7.3
• Thanksgiving
• Lecture 26 (Nov 28) Fourier Series. Laplace equation and Heat equation. note, video
• The required content from [ODE] is 2.1, 2.3.2, 2.4.1, 2.4.3, 2.5.1 and 3.6.2, 3.6.3, 3.7.3, 3.7.4 . One should focus on 2.1 and 2.4.1, and 3.6.2
• Final: Dec 14, Wed 8-11am, Cory 289