John Trangenstein: Math 107 Linear Algebra and Ordinary Differential Equations

John Trangenstein : Math 107 Linear Algebra and Ordinary Differential Equations

Class hours: MWF 10:05-11:20 Physics 047
Office hours: MWF 2:00-3:00 Physics 029C
Course grade depends on:

Homework (5%)
3 Midterms (20% each)
Final (35%)

How to get Maple software on CD from OIT for $15 First Midterm: Friday, October 3, on chapters 1 and 2
midterm1 answers
High = 100, low = 67, mean = 89, median = 91

Second Midterm: Friday, October 24 on section 2.5, section 3.6 and chapter 4
High = 96, low = 45, mean = 66, median = 70
midterm2 answers

Third Midterm: Friday, November 21
Chapter 5 and Chapter 9
midterm3 answers
High = 94, low = 41, mean = 64, median = 61

Final Exam: Friday, December 12 7-10 pm in Soc-Psych 126 Total score = midterm1 + midterm2 + midterm3
High = 283, low = 157, median = 220.5, average = 223

Syllabus:

Chapter 1: Matrices and Determinants
1.1: Systems of Linear Equations 8/25
- Maple worksheet for Gaussian elimination and back substitution
- Maple worksheet for Gauss-Jordan Elimination
- Similar Matlab commands:
  - A=[1 -1 1; 2 -3 4; -2 -1 1]
  - b=[0; -2; 7]
  - rref([A b]) % Gauss-Jordan elimination
  - x=A\b % x solves Ax=b
  - A=[2 3 -1; -1 -1 3; 1 2 2; 0 1 5]
  - b=[3; 0; 3; 3]
  - x=A\b % x minimizes Ax-b, Matlab warns user
  - A=[4 -8 -1 1 3; 5 -10 -1 2 3; 3 -6 -1 1 2]
  - b=[0; 0; 0]
  - x=A\b % x smallest solution to Ax=b, Matlab warns user
  - A=[2 1 1; 1 -2 -1; 3 -1 0; 4 -3 -1]
  - b=[0; 0; 0; 0]
  - x=A\b % x smallest that minimizes Ax-b, Matlab warns user
1.2: Matrices and Matrix Operations 8/27
- Maple LinearAlgebra worksheet for matrix multiplication
- Maple linalg worksheet for matrix multiplication
- Similar Matlab commands:
  - A=[1 2; 3 4]
  - B=[5 6; 7 8]
  - A+B
  - A*B
  - 3*A
1.3: Inverses of Matrices 9/1
- Formulas for Elementary Matrices
- Maple LinearAlgebra worksheet for matrix inverse
- Maple linalg worksheet for matrix inverse
- Similar Matlab commands:
  - A=[2 1 3; 2 1 1; 4 5 1]
  - inv(A)
  - A*inv(A)
  - B=[2 -6; -3 9]
  - inv(B) % Matlab warns user
  - B*inv(B) % Matlab returns zero matrix
1.4: Additional Properties of Matrices 9/1
- Maple LinearAlgebra worksheet for matrix transpose
- Maple linalg worksheet for matrix transpose
- Similar Matlab commands
  - A=diag([2 -4 7 3])
  - inv(A)
  - U=[-2 3 1; 0 4 -2; 0 0 5] % upper triangular
  - inv(U) % also upper triangular
  - L=[1 0 0; 2 3 0; 6 0 5] % lower triangular
  - inv(L) % also lower triangular
  - C=[1 2 3; 4 5 6]
  - C' % Matlab transpose of C
  - S=[1 2 -3; 2 -1 4; -3 4 0]
  - S' % same as S
1.5: Determinants 9/3
Determinants are Volumes
Important Facts About Determinants
Determinants computed without using expansion by minors
Maple linalg worksheet for determinants and adjoints
Maple LinearAlgebra worksheet for determinants and adjoints
1.6: Properties of Determinants 9/8
More Important Facts About Determinants
Vandermonde Determinants
1.7: Proofs of Theorems on Determinants 9/8

Chapter 2: Vector Spaces

2.3: Linear Independence and Bases 9/8
2.1: Vector Spaces 9/15
2.2: Subspaces and Spanning Sets 9/15, 9/17
Important Facts About Subspaces, ...
2.3: Linear Independence and Bases 9/17
2.4: Dimension, Nullspace, Row Space and Column Space 9/22
2.5: Wronskians 9/24

Chapter 3: First Order Ordinary Differential Equations

3.6: Modeling with Differential Equations 9/24

Chapter 4: Linear Differential Equations
Maple worksheet for solving ordinary differential equations

4.1 Higher Order Linear ODEs 9/29
4.2 Homogeneous Constant Coefficient Linear ODEs 10/1
4.3 Undetermined Coefficients 10/6 Undetermined Coefficients without Guessing
4.4 Variation of Parameters 10/6
Variation of Parameters without Wron(g)skians
4.5 Applications 10/8
Particular Solutions of Mass-Spring-Dashpot Equations with Periodic Forcing

Chapter 5: Linear Transformations and Eigenvectors

5.1: Linear Transformations 10/20
5.2: Differential Operators and Differential Equations 10/20
5.3: Matrices for Linear Transformations 10/22 Matrix Representations of Rotations, Reflections and Projections
5.4: Eigenvalues and Eigenvectors 10/27
Facts about Eigenvalues
Maple worksheet for eigenvalues
5.5: Similar Matrices, Diagonalization and Jordan Forms 10/29
Maple worksheet for Jordan canonical form

Chapter 9: Inner Product Spaces

9.1: Inner Product Spaces 11/3
9.2: Orthonormal Bases 11/3
9.3: Schur's Theorem and Symmetric Matrices 11/5
Deflation, Schur Decomposition, Symmetric Matrices, Singular Value Decomposition

Chapter 6: Systems of Differential Equations

6.1: Theory 11/10
6.5: Conversion to First Order Systems 11/10
6.2: Diagonalizable Homogeneous Systems 11/12
Matrix Exponentials for Solving Systems of Linear Differential Equations
6.3: Nondiagonalizable Homogeneous Systems 11/17
6.4: Nonhomogeneous Systems 11/19
6.6: Applications 11/24
6.7: Systems of 2 Nonlinear Differential Equations 12/1

Homework:

due September 1
- 1.1 (p 15): 2,8,15,17,19,22,23,26,28,30
- 1.2 (p 26): 5,9,11,12,14,18,20,21,23,28,29,30,37
due September 8
- 1.3 (p 36): 1,6,7,10,11(b),13,14,16,26
- 1.4 (p 41): 4,12,17,20,22(c),24(d),26,32,33
  An easy way to prove problem 33 for lower triangular matrices is to multiply the matrix times any axis vector e_j, and show that the result involves axis vectors e_{j+1}...e_n. If you do this n times, the final answer must be zero. A similar statement applies to upper triangular matrices.
- 1.5 (p 50): 5,8,12,15,16
due September 15
- 1.6 (p 57): 4,6,10,13,15(c),16
- 1.7: 5,6,7
- 2.3: 1,2,3,6
due September 22
- 2.1 (p 73): 3,6
- 2.2 (p 81): 1(c)(d),2(b)(d),3(c),5,11,12,13,21,22
- 2.3 (p 93): 7,10,14,17,21,24,27,28
- 2.4 (p 104): 2,3(a)(b),4(c)(d),7,10,14,18,26
due September 29
- 2.5 (p 110): 5,6,12,14,16
- 3.6 (p 151): 1,2,4,11,13,15,16
- 4.1 (p 188): 2,3,6,10,11,15,17,24
October 6
- 4.2 (p 201): 2,5,7,10,11,12,13,20,22,23,24,29,30,40,41,42
- 4.3 (p 211): 1,4,9,11,18,26
- 4.4 (p 217): 1, 2, 3
October 20
- 4.5 (p 228): 1,3,5,6,7,8,11,13,15,16
- 5.1 (p 243): 3,4,7,12,18,20,33,35,36
October 27
- 5.2 (p 252): 5,6,8,11,14,17,20,23(a,b)
- 5.3 (p 265): 1(a--d), 4(a--d)
November 3
- 5.4 (p 277): 5,8,9,16,17,20,26,32(is this matrix diagonalizable?)
- 5.5 (p 286): 5,8,9,16,17,21,24,30,31,36,40
November 10
- 9.1 (p 419): 6,7,8,16,18,20,21,22
- 9.2 (p 429): 2,6,9,13
- 9.3
November 17
- 6.1 (p 301): 1,4,5,9,17,27,28
- 6.5 (p 322): 4,5,13
- 6.2 (p 311): 3,5,11,15,22,25,28,30
November 24
- 6.3 (p 314): 3,17
- 6.4 (p 318): 1,5,11,13,15
December 1
- 6.6 (p 331): 2,4,11,15
- 6.7

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