Teaching Linear Algebra at UNSW Sydney

Transcription

1 Teaching Linear Algebra at UNSW Sydney Jie Du University of New South Wales via Xiamen Unniversity 30 December, 2017 Jie Du Teaching LA at UNSW 30 December, / 15

2 Jie Du Teaching LA at UNSW 30 December, / 15

3 Jie Du Teaching LA at UNSW 30 December, / 15

4 Jie Du Teaching LA at UNSW 30 December, / 15

5 Jie Du Teaching LA at UNSW 30 December, / 15

6 Jie Du Teaching LA at UNSW 30 December, / 15

7 Jie Du Teaching LA at UNSW 30 December, / 15

8 The School consists of three departments: 1 Applied Mathematics 2 Pure Mathematics 3 Statistics 4 The school is run by the committees Policy and Resource Committee (Heads of School, Departments, and Committees, PGC, Director of First Year Studies, Professors) Standing Committee (Representatives of every level, Heads, etc.) Research Committee Academic Committee Computing, Publicity,... Jie Du Teaching LA at UNSW 30 December, / 15

9 Research Groups 1 In Pure Maths: Algebra and Number Theory Combinatorics Functional and Harmonic Analysis Geometry Mathematical Physics 2 In Applied Maths: Biomathematics Computational Mathematics Fluid Dynamics, Oceanic and Atmospheric Sciences Nonlinear Phenomena Optimization 3 In Statistics: Bayesian and Monte Carlo Methods Biostatistics and Ecology Finance and Risk Analysis Nonparametric Statistics Stochastic Analysis Jie Du Teaching LA at UNSW 30 December, / 15

10 Undergraduate courses 1 Junior level: Provide service teaching for the entire university. There are courses specially designed for engineering, computing science, Banking, Finance and Actuarial Studies, etc. Jie Du Teaching LA at UNSW 30 December, / 15

11 Undergraduate courses 1 Junior level: Provide service teaching for the entire university. There are courses specially designed for engineering, computing science, Banking, Finance and Actuarial Studies, etc. 2 Senior level: High Algebra, High analysis, courses for Honours students, etc. Jie Du Teaching LA at UNSW 30 December, / 15

12 Linear Algebra Courses 1 First year linear algebra Semester 1: points, lines, planes in R n, vector geometry, dot product, orthogonal projection, cross product, distance, linear equations and matrices, matrix operations. Jie Du Teaching LA at UNSW 30 December, / 15

13 Linear Algebra Courses 1 First year linear algebra Semester 1: points, lines, planes in R n, vector geometry, dot product, orthogonal projection, cross product, distance, linear equations and matrices, matrix operations. 2 First year linear algebra Semester 2: Vector space, linear transformations, eigenvalue and eigenvectors. Jie Du Teaching LA at UNSW 30 December, / 15

14 Linear Algebra Courses 1 First year linear algebra Semester 1: points, lines, planes in R n, vector geometry, dot product, orthogonal projection, cross product, distance, linear equations and matrices, matrix operations. 2 First year linear algebra Semester 2: Vector space, linear transformations, eigenvalue and eigenvectors. Applications: Markov process, Input output model, diagonalization and power of matrices. Jie Du Teaching LA at UNSW 30 December, / 15

15 Linear Algebra Courses 1 First year linear algebra Semester 1: points, lines, planes in R n, vector geometry, dot product, orthogonal projection, cross product, distance, linear equations and matrices, matrix operations. 2 First year linear algebra Semester 2: Vector space, linear transformations, eigenvalue and eigenvectors. Applications: Markov process, Input output model, diagonalization and power of matrices. 3 Second year linear algebra Semester 1:... LU factorisation, Symmetric matrix, Inner product space, JCF. Jie Du Teaching LA at UNSW 30 December, / 15

16 Linear Algebra Courses 1 First year linear algebra Semester 1: points, lines, planes in R n, vector geometry, dot product, orthogonal projection, cross product, distance, linear equations and matrices, matrix operations. 2 First year linear algebra Semester 2: Vector space, linear transformations, eigenvalue and eigenvectors. Applications: Markov process, Input output model, diagonalization and power of matrices. 3 Second year linear algebra Semester 1:... LU factorisation, Symmetric matrix, Inner product space, JCF. Applications: System of first order ODEs, exponential of matrices, matrix groups, etc. There are ordinary and higher classes. Jie Du Teaching LA at UNSW 30 December, / 15

17 Linear Algebra Courses 1 First year linear algebra Semester 1: points, lines, planes in R n, vector geometry, dot product, orthogonal projection, cross product, distance, linear equations and matrices, matrix operations. 2 First year linear algebra Semester 2: Vector space, linear transformations, eigenvalue and eigenvectors. Applications: Markov process, Input output model, diagonalization and power of matrices. 3 Second year linear algebra Semester 1:... LU factorisation, Symmetric matrix, Inner product space, JCF. Applications: System of first order ODEs, exponential of matrices, matrix groups, etc. There are ordinary and higher classes. 4 Geometries and transformation groups Semester 2 Jie Du Teaching LA at UNSW 30 December, / 15

18 Linear Algebra Courses 1 First year linear algebra Semester 1: points, lines, planes in R n, vector geometry, dot product, orthogonal projection, cross product, distance, linear equations and matrices, matrix operations. 2 First year linear algebra Semester 2: Vector space, linear transformations, eigenvalue and eigenvectors. Applications: Markov process, Input output model, diagonalization and power of matrices. 3 Second year linear algebra Semester 1:... LU factorisation, Symmetric matrix, Inner product space, JCF. Applications: System of first order ODEs, exponential of matrices, matrix groups, etc. There are ordinary and higher classes. 4 Geometries and transformation groups Semester 2 Jie Du Teaching LA at UNSW 30 December, / 15

19 Teaching in English and Linear Algebra teaching Challenges: Jie Du Teaching LA at UNSW 30 December, / 15

20 Teaching in English and Linear Algebra teaching Challenges: 1 Challenge 1: Teaching in English. At least two year practice before teaching confidently. Jie Du Teaching LA at UNSW 30 December, / 15

21 Teaching in English and Linear Algebra teaching Challenges: 1 Challenge 1: Teaching in English. At least two year practice before teaching confidently. 2 Challenge 2: Research Oriented Teaching. Jie Du Teaching LA at UNSW 30 December, / 15

22 Teaching in English and Linear Algebra teaching Challenges: 1 Challenge 1: Teaching in English. At least two year practice before teaching confidently. 2 Challenge 2: Research Oriented Teaching. 3 Challenge 3: Shift your teaching methods among various level of students. Example 1 Jie Du Teaching LA at UNSW 30 December, / 15

23 Teaching in English and Linear Algebra teaching Challenges: 1 Challenge 1: Teaching in English. At least two year practice before teaching confidently. 2 Challenge 2: Research Oriented Teaching. 3 Challenge 3: Shift your teaching methods among various level of students. Example 1 Professional Teachers: Jie Du Teaching LA at UNSW 30 December, / 15

24 Teaching in English and Linear Algebra teaching Challenges: 1 Challenge 1: Teaching in English. At least two year practice before teaching confidently. 2 Challenge 2: Research Oriented Teaching. 3 Challenge 3: Shift your teaching methods among various level of students. Example 1 Professional Teachers: 1 Your passion Jie Du Teaching LA at UNSW 30 December, / 15

25 Teaching in English and Linear Algebra teaching Challenges: 1 Challenge 1: Teaching in English. At least two year practice before teaching confidently. 2 Challenge 2: Research Oriented Teaching. 3 Challenge 3: Shift your teaching methods among various level of students. Example 1 Professional Teachers: 1 Your passion 2 Thoroughly prepare each lecture: three year cycles Jie Du Teaching LA at UNSW 30 December, / 15

26 Teaching in English and Linear Algebra teaching Challenges: 1 Challenge 1: Teaching in English. At least two year practice before teaching confidently. 2 Challenge 2: Research Oriented Teaching. 3 Challenge 3: Shift your teaching methods among various level of students. Example 1 Professional Teachers: 1 Your passion 2 Thoroughly prepare each lecture: three year cycles 3 The first five minutes Jie Du Teaching LA at UNSW 30 December, / 15

27 Teaching in English and Linear Algebra teaching Challenges: 1 Challenge 1: Teaching in English. At least two year practice before teaching confidently. 2 Challenge 2: Research Oriented Teaching. 3 Challenge 3: Shift your teaching methods among various level of students. Example 1 Professional Teachers: 1 Your passion 2 Thoroughly prepare each lecture: three year cycles 3 The first five minutes 4 Handling students questions. Example 2 Jie Du Teaching LA at UNSW 30 December, / 15

28 Teaching in English and Linear Algebra teaching Challenges: 1 Challenge 1: Teaching in English. At least two year practice before teaching confidently. 2 Challenge 2: Research Oriented Teaching. 3 Challenge 3: Shift your teaching methods among various level of students. Example 1 Professional Teachers: 1 Your passion 2 Thoroughly prepare each lecture: three year cycles 3 The first five minutes 4 Handling students questions. Example 2 5 High tech facilities/on-line teaching Jie Du Teaching LA at UNSW 30 December, / 15

29 Example 1 Random Variables Let F be a field of 5 elements and V = F 2, a space of 25 vectors. Two vectors v 1, v 2 are chosen in order from V at random. This has sample space S = V V. Define a random variable Z on S by setting Z(v 1, v 2 ) = dim span{v 1, v 2 }. Find the probability distribution of Z. Solution. 1 P(Z = 0) = =.16%; 2 P(Z = 2) = = 76.8%: v 1 can be any vector in F 2 0, and v 2 must be in F 2 Fv 1 ; 3 P(Z = 1) = = = 23.04%. Jie Du Teaching LA at UNSW 30 December, / 15

30 Example 2 Transformation Groups Theorem A similarity without a fixed point is an isometry. Proof. First, any similarity α = τδ 0,r for some r > 0 and isometry τ. So α has equation α(x) = rqx + b for all x R n, where Q is an orthogonal matrix and b R n. Suppose now that α has no fixed point. In other words, the linear system (rq I )x = b has no a solution. This implies det(rq I ) = 0 or equivalently, det(q r 1 I ) = 0. Hence, r 1 is an eigenvalue of Q. Suppose v is an associated eigenvector, i.e., Qv = r 1 v. Since Q is orthogonal, we have v and r 1 v have the same length, forcing r = 1. Hence, α = τ is an isometry. Jie Du Teaching LA at UNSW 30 December, / 15

31 THANK YOU! Jie Du Teaching LA at UNSW 30 December, / 15