Lecture Notes

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====== Math 54 Honor (2022 Fall)======
Course Number 24291 \\

Lecture: \\
TuTh 11:00A-12:29P - 289 - Cory\\
Instructor: Peng Zhou, pzhou.math@gmail.com \\
Zoom: [[https://berkeley.zoom.us/j/92643467269?pwd=MVJGd3FNaTA5dmFFSHdwYnBXV2lTZz09|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 =====
  * [[https://discord.gg/K2rDuW8USM | Discord server]] I will keep an eye on it and answering questions. (There is also an unofficial one around.)
  * [[https://stackoverflow.com/c/math-h54/questions | Stack Overflow for Math H54]] This is the team version. You can register with berkeley email
  * [[https://edstem.org/us/join/Gt5e8Z | Ed Discussion ]] (hmm, something new, let's try it out)
  * [[.:s: |Student Area]]: you can post your lecture notes and homework solutions, as a reference to your fellow class mates. 
  * [[https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab | 3Blue1Brown]] You can visualize stuff.  
  * Wolfram-Alpha: https://www.wolframalpha.com , online graphing tool, and also good for computing
  * [[https://www.overleaf.com | Overleaf]]. If you want to latex your homework, try this one. 

===== Grading =====
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 <del>Monday</del> 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%. 




===== Reference =====

  * Givental. [LA] [[https://math.berkeley.edu/~giventh/la.pdf | Linear Algebra]], and [ODE]  [[https://math.berkeley.edu/~giventh/papers/ode.pdf | Ordinary Differential Equation]]
  * Peterson [[https://link.springer.com/book/10.1007/978-1-4614-3612-6 | Linear Algebra ]]

[[proof|about writing proof ]]
 
===== 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): {{ :math54-f22:math_h54_quiz_1_no_solutions.pdf |quiz 1}}, {{ :math54-f22:math_h54_quiz_1.pdf |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)
  * [[hw2-quiz|sample quiz 2 questions]]
  * Lecture 4 (Sep 6): [ODE 1.3]. 2x2 matrices: determinant, inverse. Orthogonal transformation.[[https://berkeley.zoom.us/rec/share/XEuQ6eo0CrZnmZ2iP2vhXWY70071XyVgbahch_wA80t1YNmo21OhbZbf64QOUMk.AZT9P0X6IifG6oH3 | video ]] 
  * {{ :math54-f22:math_h54_quiz_2.pdf |quiz 2}}, {{ :math54-f22:math_h54_quiz_2_solutions.pdf |solution}}
  * Lecture 5 (Sep 8): [LA] Ch 1 Sec 3. Orthogonal Transformations. Complex Numbers. [[https://berkeley.zoom.us/rec/share/UMC0f8ckUAY4HHgoZamwcQRnmESGAP-sRBSVjXkWj1h3Ac9L_J0CCQjQBKS6Dng6.aae13Hwkh_qtLi8u | 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) [[https://berkeley.zoom.us/rec/share/fE4C6R0ArPqotUMUOPQXeJXMp53MqVVOALfJr2r2JyaNU6VVh38DVtyXoqI6wuY.0wWp110xJ0Vwtvfc | video]]
  * Lecture 7 (Sep 15) [LA] 2.1 Matrices, 2.2 Determinants [[ https://berkeley.zoom.us/rec/share/yMpHmsa-1x3YjZErHeicrBiVsZIAK_gI-5AaVQpGcGo7ScwrwqbvlQUaZ6uEO6g7.VBXiz0N_Lc9sZj-C | 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), $\det(AB)= \det(A) \det(B)$.  [[https://berkeley.zoom.us/rec/share/4pYG46VhcKGVPG7Uk_dBjY4APXXk0gFetuRU06Q8BkcrnwZTl1tkn1fuGl3YFHvH.bLTsnXc82-ayui4Y|video]]
  * Lecture 9 (Sep 22) [LA] finish up 2.2 about cofactor and block matrix multiplication [[https://berkeley.zoom.us/rec/share/sPrYQSlRZCkQXuAaHx1fGHIG2myJwmP3XuKqcygg-JzIT1Ggjdku34z-8Bx-bHaM.0M3ZRHh6EMS6LuqG | video ]]
  * HW 5: 110, 118, 132, 135, 138, 139, 140, 143
  * [[quiz3|sample quiz 3 questions]]
  * [[quiz4|sample quiz 4 questions]]
  * Lecture 9 (Sep 27) [LA] 2.3 Abstract Vector spaces. Fields. [[https://berkeley.zoom.us/rec/share/P-uibRoS4OJNw12j6NQIaE2ntkX-DiBKaNXsPui7jNVn90AIbbr35Xz-2lpaZNuc.AfnO1b8a0Z9ECmgt|video]]
  * Lecture 10 (Sep 29) [LA] 3.1 Dimension and Ranks [[https://berkeley.zoom.us/rec/share/t8NeevBtQZ-rHgyh3DkAlLmJL1sUt94rfRQoobKBKYSp5V656EB96oh4R99aXqqo.Cnj-SdhaUpV2HWtB | video]]
  * HW 6: 150,151,152,155,159(notice $W^\perp$ in general lives in the dual space),160, 165, 167
  * [[quiz5|sample quiz 5 questions]]
  * [[[[math54-f22:midterm1-sample|]]
  * Lecture 11 (Oct 4) [LA] 3.1. rank theorem [[https://berkeley.zoom.us/rec/share/1gWfwQPsh2QgYuexLPuib4YBsBRxhUAToW9SOI3VeG5-Xq_Nwo8QPtUn7MFtWqrW.wzw6z5vSD6aW5MsN| 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) [[https://berkeley.zoom.us/rec/share/2vhx5icZ0K3lqCChrgGlZvS11xWawLMgP4P1r45_nVvxfv_Jg0DbPUOj3T0A4d8C.D5eMYxSpCVQvaLUj | 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 [[https://berkeley.zoom.us/rec/share/28a1j-JXfAXemhPUdJ6VD7c_lNcqWjrKmKgayBTa7DdTJDJb5RotPZJRxEWAavhV.dzF9_dH4B2W3r8JI | video]]
  * Post Midterm 1 [[https://forms.gle/3xJhw5J4zGyEWLAfA|Feedback form(anonymous)]]
  * Lecture 14 (Oct 13) [LA] Q&A on dual vector space, dual basis and quotient space. [[https://berkeley.zoom.us/rec/share/v9G1c6mA5coSr0lT3RHub2TEQKtuas7SlCabYJH1Ahs9VQclmQiQDgIxVRQqE71J.QnxL877EbiQBYvCR | video]]
  * [[quiz6|sample quiz 6 questions]]
  * HW 8: 
     * [LA] read about LPU decomposition. give some 2x2 and 3x3 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 variety[[https://berkeley.zoom.us/rec/share/v8yjWMev5mDH0_1sDugHBnIr9c6rUfU_SeZ4a-VppUjiPwW-HNyOqr5IJJ7GI80w.YDi8kYAWTjTl6N5U|video]]
  * [[quiz7|sample quiz 7 questions]]
  * Lecture 16 (Oct 20) [LA] 3.3 Quadratic form [[https://berkeley.zoom.us/rec/share/7Oku10cOWRnKW96oxyD1h5otBChzeTviZDnfxP0AYh4djpVe9K0iB1yzcdVQipL2.BdUsUjMvVx3MhbO0 | 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. [[https://berkeley.zoom.us/rec/share/MXntTgJzX7c2_O6EJLBwegGNfeSA7ItreWrGkOyGmXd3bvD6JLE8lWgdAifsUNl0.QXRzIVysKzY-44NT | video ]]
  * Lecture 18 (Oct 27): [LA] Sylvester theorem. Orthogonalization preserving flag. [[https://berkeley.zoom.us/rec/share/wnIHm7f8FzX-94PiIvE3uldOFwd31dd5UTqo1H-2cv1IG-9f90bY2XK6EJijVCYZ.emZA1iws-YrlQKDa | video ]]
  * HW 10: 226, 229,230, 232, 233, 234, *239
  * [[quiz8|sample quiz 8 questions]]
  * [[quiz9|sample quiz 9 questions]]
  * Lecture 19 (Nov 1): [LA] Cauchy-Schwarz. Adjoint map. Normal Operator. [[https://berkeley.zoom.us/rec/share/f92YzkRy3l92cUcvvDeXWqK6YXlJ3S9AsbsHgzCAd74clGcMWK-eT3f3XmZ9zXBN.hmAynflyfaSvWRQo|video]]
  * Lecture 20 (Nov 3): [LA] Proof of Spectral theorem for normal operator. Complexification of real vector spaces. [[ https://berkeley.zoom.us/rec/share/aYtqrb4dAviwr7bR38XCgXeq2PIT7af_8EQTkRGReFuGj-AlK0K1-87XIjOZuCyx.pSoaTX7LBmBkY3MN | video]]
  * HW 10: 253-257, 260, 261, 263,265
  * Lecture 21 (Nov 8) [ODE] 2.1.1 and 2.1.2 [[https://berkeley.zoom.us/rec/share/3P0S1EI86XI2bIp6Cnwajh0QvdD7ZtOHIGPIIWbbaxR7zEXZkkD-SqZpmYMWnqrB.C6GXYwXpaY7vPRlL| video]]
  * No quiz on Nov 9 (wednesday)
  * Lecture 22 (Nov 10) [ODE] 2.1.2 and [LA] 4.2 Jordan form [[https://berkeley.zoom.us/rec/share/yOJRpg4mmzmwhWK_J_SH7qDxhT8ajLr9-9ttMgzuxFauYkJotxrPawMhVSi02I48.wFNomaEoKJkZFjxZ | video]]
  * [[Sample Midterm 2]]
  * Midterm 2. Nov 14 Monday in discussion. [[midterm 2|more info here]], [[midterm 2 changed to online]]
  * {{ :math54-f22:midterm-2.pdf | Midterm 2 Exam}}
  * Lecture 23 (Nov 15) Jordan form [[https://berkeley.zoom.us/rec/share/JAZ_IOwu1K22tK7bUl9iFD17W1ZVMHsQLXUz4xW8GuFTrQhWN0zfQ_O7MYPyPH4h.GjInO7ZV_o8bGwbc| video]], {{ :math54-f22:lecture_23.pdf | note}}
  * Lecture 24 (Nov 17) Constant Coeff Equation. {{ :math54-f22:lecture_24.pdf | note }}, [[https://berkeley.zoom.us/rec/share/vDvN04tnjgzl9Ah_-niISUp9IPkjzHRVe1CYhxyHBytmwdniJ-88dtyl1laCMur5.ULD111wRqI_A0XnC | 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. {{ :math54-f22:lecture_25.pdf | note}}, [[https://berkeley.zoom.us/rec/share/wu-tg6u_lw8H3vpQElzCB5UztEzydvjYrmofqv7B9NigniOgYr5io_r4Muv_gjaE.UwCqC6ffq4wtfa1U | video ]]
  * Midterm 2 Discussion (Nov 22 11:10-12:30). {{ :math54-f22:midterm-2_solution.pdf |note}}, [[https://berkeley.zoom.us/rec/share/P5FxvJ4ceAc_64nLsshnmad4L_hLEtwwINbUD_tx9oZlNH2mxKo2BBOXnC3n-tc_.U2o2hAvsvjBnCZ1d | video]]
  * HW: [ODE] p128, Ex 3.7.3
  * Thanksgiving
  * Lecture 26 (Nov 28) Fourier Series. Laplace equation and Heat equation. {{ :math54-f22:lecture_26.pdf |note}}, [[https://berkeley.zoom.us/rec/share/X-1rQgVV3TVJSY3b-k5PhwP_WJi_PoF9qsN0GI-ESO8DahH_TPKHeKEc8wwMvJz7.DK9RngVFlfIDxG03 | video]]


  * Lecture 27 (Dec 1) 
       * Review of DE (30 min) {{ :math54-f22:lecture_27-1.pdf | note}}, [[https://berkeley.zoom.us/rec/share/XnSK3xYFgPhWvjUv-e_bBu6LIcZp2chpsumwAs6FGIx5RF8hL87ylsbHpPi5-ifo.l2OhPRLX6FBruH39 | video]]
       * Going over past final. {{ :math54-f22:final-2014f.pdf | 2014 fall final}}, [[https://berkeley.zoom.us/rec/share/rfiSx4IHeag4l13mayrAdMYJBs4j6oNIl_gBX6VschmS3c1Dvbj4rmV21OWGjos9.oF2dLbqdDFTxYWA3 | video]]
       * more finals can be [[https://math.berkeley.edu/~peyam/Math54Sp15.html | found here]]. You can try [[https://math.berkeley.edu/~peyam/Math54Fa11/Practice%20Exams/Final%20(Canez).pdf | Final (Canez)]] first.

  * 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
===== Previous Quizzes and Solutions =====
  * {{ :math54-f22:math_h54_quiz_1_no_solutions.pdf |quiz 1}}, {{ :math54-f22:math_h54_quiz_1.pdf |solution}}
  * {{ :math54-f22:math_h54_quiz_2.pdf |quiz 2}}, {{ :math54-f22:math_h54_quiz_2_solutions.pdf |solution}}
  * {{ :math54-f22:math_h54_quiz_3.pdf |quiz 3}}, {{ :math54-f22:math_h54_quiz_3_solutions.pdf |solution}}
  * {{ :math54-f22:math_h54_quiz_4.pdf |quiz 4}}, {{ :math54-f22:math_h54_quiz_4_solutions.pdf |solution}}
  * {{ :math54-f22:math_h54_quiz_5.pdf |quiz 5}}, {{ :math54-f22:math_h54_quiz_5_solutions.pdf |solution}}
  * {{ :math54-f22:math_h54_quiz_6.pdf |quiz 6}}, {{ :math54-f22:math_h54_quiz_6_solutions.pdf |solution}}
  * {{ :math54-f22:math_h54_quiz_7.pdf |quiz 7}}, {{ :math54-f22:math_h54_quiz_7_solutions.pdf |solution}}
  * {{ :math54-f22:math_h54_quiz_8.pdf |quiz 8}}, {{ :math54-f22:math_h54_quiz_8_solutions.pdf |solution}}
  * {{ :math54-f22:math_h54_quiz_9.pdf |quiz 9}}, {{ :math54-f22:math_h54_quiz_9_solutions.pdf |solution}}
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