https://courses.pzhou.org/ 2026-04-17T08:36:08+00:00 https://courses.pzhou.org/ https://courses.pzhou.org/lib/tpl/dokuwiki/images/favicon.ico text/html 2026-02-21T14:41:16+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math54-f22:hw2-quiz&rev=1771684876&do=diff quiz-style questions: * Write down a linear transformation that takes $x^2/4+y^2=1$ to the unit circle. Is it unique? (which means: Is there more than one such linear transformation? ) * Write down a linear transformation that takes the line $x=y$ to $x=0$. Is it unique? text/html 2026-02-21T14:41:16+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math54-f22:midterm1-sample&rev=1771684876&do=diff Sample Midterm 1 Our midterm will be 50 minutes, hold in discussion session on next Wednesday (Oct 5). The test range is Givental [LA] Ch 1 and Ch 2, and Ch 3 with the concept of basis, linear independence and span. We will have 5 problems, each worth 20 points. text/html 2026-02-21T14:41:16+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math54-f22:midterm_1&rev=1771684876&do=diff Midterm 1 [ exam], [solution] Stat Range Count 90 - 100 14 80 - 90 7 70 - 80 6 60 - 70 6 50 - 60 4 Total: 37 Average: 80 text/html 2026-02-21T14:41:16+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math54-f22:midterm_2&rev=1771684876&do=diff Midterm 2 We will cover Chapter 3 and Chapter 4.1, eigenvalue problem. Key concepts and skills * Given a linear map $T: V \to V'$, what are $\ker(T), im(T)$, what is $rank(T)$? * If $W \In V$ is a subspace, then what's the relation between dimension of $W$, $V/W$ and $V$? text/html 2026-02-21T14:41:16+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math54-f22:midterm_2_changed_to_online&rev=1771684876&do=diff To prepare for the UAW strike this coming week, we will move the midterm online this time. The exam will be released on the course website on Monday 11:00am, and you can email me your scanned exam by 12:10noon. My email is pzhou.math@berkeley.edu . In the email subject line, please use [Math H54 Midterm 2] - Name - student ID. Please scan into a single pdf file. You can also write on ipad and send me the pdf. text/html 2026-02-21T14:41:16+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math54-f22:proof&rev=1771684876&do=diff Hi, I know many of you have not written proofs before, although our course is not proof-based, we do want us to get into the habit of giving reasons. This a natural thing when you are talking to someone. You say, 'I like orange', they ask 'why?', you say, 'I don't know, I just do'. That's fine in life, since many things cannot be explained. That would be funny when you do math, where math is one of the few lucky places where reasoning works. text/html 2026-02-21T14:41:16+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math54-f22:quiz3&rev=1771684876&do=diff quiz-style questions: * Find expressions for $\sin(2\theta)$ and $\sin(3\theta)$ in terms of $\cos(\theta), \sin(\theta)$. * Compute $(\frac{1+i}{\sqrt{2}})^{20}$ * Sketch the region $\vert z-1 \vert + \vert z + 1 \vert \leq 3$. text/html 2026-02-21T14:41:16+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math54-f22:quiz4&rev=1771684876&do=diff quiz-style questions: * Find the sign of the permutation $(4312)$. * Compute $\det(A - \lambda I)$ where $$A = \begin{bmatrix} 1 & 0 & -1\cr 0 & 1 & -1\cr -1 & 1 & 0 \end{bmatrix}$$ * Let $\sigma$ be a permutation of length $\ell > 0$. Show that there must exist some $i$ such that $\sigma(i) > \sigma(i+1)$. text/html 2026-02-21T14:41:16+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math54-f22:quiz5&rev=1771684876&do=diff quiz-style questions: * Compute $$\begin{vmatrix} 1 & x & x^2 \cr x & 1 & x^2 \cr x^2 & x & 1 \end{vmatrix}$$ * Prove that in $\mathbb{K}[x]$ polynomials of degree $n$ do not form a subspace, but polynomials of degree $\leq n$ do. * Find a linear transformation $T: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ such that the volume of the 3-D parallelogram (parallelopiped) defined by $T(1,0,0), T(0,1,0),$ and $T(0,0,1)$ is $\frac{1}{2}$. text/html 2026-02-21T14:41:16+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math54-f22:quiz6&rev=1771684876&do=diff Sample Quiz 6 Questions * If $A$ is $n \times n$ and $Ax=0$ for some $x \neq 0$, can $A$ be of full rank? Why? What is $det(A)$? * Prove that every subspace in $\mathbb{K}^n$ can be described as the range of a suitable linear map. Prove that every subspace of $\mathbb{K}^n$ can be described as the kernel of a linear map. text/html 2026-02-21T14:41:16+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math54-f22:quiz7&rev=1771684876&do=diff Sample Quiz 7 Questions * What is $A^{-1}$ if $$A = \begin{bmatrix} 1 & 0 & 1 \cr 0 & 1 & 1 \cr 0 & 0 & 1 \end{bmatrix}$$ * Write $$B = \begin{bmatrix} 1 & 2 & 3 \cr 4 & 5 & 6 \cr 7 & 8 & 9 \end{bmatrix}$$ in row-echelon form. * What are the image and kernel of the matrix $$C = \begin{bmatrix} 1 & 1 & 1 \cr 1 & 2 & 1 \cr 2 & 3 & 2 \end{bmatrix},$$ text/html 2026-02-21T14:41:15+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math54-f22:quiz8&rev=1771684875&do=diff Sample Quiz 8 Questions * LPU decompose the following matrix: $$M = \begin{bmatrix} 1 & 2 & 3 \cr 0 & 1 & 2 \cr 1 & 2 & 9 \end{bmatrix}$$ * Let $B$ be a symmetric 3×3 matrix (pick one yourself), and find an orthogonal basis for $B$. text/html 2026-02-21T14:41:16+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math54-f22:quiz9&rev=1771684876&do=diff Sample Quiz 9 Questions True or False: Please provide brief justification * A real symmetric matrix is Hermitian. * A pure imaginary matrix that is anti-symmetric is Hermitian (i.e. all coefficients are imaginary). * If $H$ is an $n \times n$ Hermitian matrix, then $\overline{v^t}Hv$ is real. text/html 2026-02-21T14:41:16+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math54-f22:sample_midterm_2&rev=1771684876&do=diff Sample Midterm 2 Problems Computational 1. Consider the following Hermitian 2×2 matrices $Q$, find an invertible matrix $A$ and diagonal matrices $D$, such that $Q = A^* D A$. * $ Q = \begin{bmatrix} 1 & 2 \cr 2 & 2 \end{bmatrix} $ * $ Q = \begin{bmatrix} 0 & i \cr -i & 0 \end{bmatrix} $ text/html 2026-02-21T14:41:16+00:00 Anonymous (anonymous@undisclosed.example.com) https://courses.pzhou.org/doku.php?id=math54-f22:start&rev=1771684876&do=diff 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