# Math 21b - Homework

Course preceptor: Yu-Wen Hsu (yuwenhsu@g.harvard.edu)
Here is the homework policy in a nutshell:
• Homework must be turned in to your CA at the start of section. If you miss a class, you are responsible for getting it to your CA on time. (It's a good idea to let your CA know how you intend to do this, to avoid miscommunications.)
• Late homework is never accepted.
• Your lowest homework score will be dropped at the end of the semester. (You can earn additional homework drops by taking quizzes in problem session.)
• You are welcome to work with other students. However, you should only turn in work that you understand yourself, and that you have written up yourself and explained in your own words. Any help you receive must be acknowledged.

Hour and topic Assignment Relevant reading Due date Solutions
0. Vector and Matrix Basics Problem Set 0 Basics of Vectors and Matrices W 9/6 Solutions
1. Introduction to Linear Systems Problem Set 1 Bretscher 1.1, The Method of Elimination F 9/8 Solutions
Weekly Problems
2. Gauss-Jordan Elimination Problem Set 2 Bretscher 1.2 M 9/11 Solutions
3. Introduction to Linear Transformations Problem Set 3 Introduction to Linear Transformations W 9/13 Solutions
4. How much data do you need to determine a linear transformation? Problem Set 4 Linear Combinations and Linear Transformations F 9/15 Solutions
5. More Examples of Linear Transformations Problem Set 5 Bretscher 2.2 (you may skip the formulas for projections and reflections involving dot products) M 9/18 Solutions
6. More on Bases of $$\mathbb{R}^n$$, Matrix Products Problem Set 6 Bretscher 2.3; you may skip the discussion of block matrices W 9/20 Solutions
7. Matrix Inverses Problem Set 7 Bretscher 2.4; you may skip the part on block matrices F 9/22 Solutions
8. Coordinates Problem Set 8 Coordinates M 9/25 Solutions
9. Image and Kernel of a Linear Transformation, Introduction to Linear Independence Problem Set 9 Bretscher 3.1 but don't worry about the term "rank" yet; we'll talk about that next week W 9/27 Solutions
10. Subspaces of $$\mathbb{R}^n$$, Bases and Linear Independence Problem Set 10
If you'd like more guidance on writing arguments showing that a set is closed under addition or scalar multiplication, check out the worksheet solutions, especially #3.
Bretscher 3.2 F 9/29 Solutions
11. Dimension and the Rank-Nullity Theorem Problem Set 11 Bretscher 3.3 M 10/2 Solutions
12. Orthogonal Projections and Orthonormal Bases Problem Set 12 Bretscher 5.1 F 10/6 Solutions
13. Determinants No problem set! (This material is covered in Problem Set 15.) Computing Determinants Using Minors; Bretscher 6.1 through Example 4; Bretscher 6.2: Theorem 6.2.1, Theorem 6.2.6, Theorem 6.2.8, Example 6
14. The Gram-Schmidt Process, The Transpose of a Matrix Problem Set 14 Bretscher 5.2 (skip QR factorization) and the following parts of Bretscher 5.3: from Definition 5.3.5 to Theorem 5.3.6, and Theorem 5.3.9 W 10/11 Solutions
15. Least Squares and Data Fitting Problem Set 15 Bretscher 5.4 through Theorem 5.4.5 only F 10/13 Solutions
16. Introduction to Discrete Dynamical Systems and Eigenanalysis Problem Set 16 §7.1 from the 4th edition of Bretscher (if you have the 5th edition, we've posted a copy here); Complex Numbers (and solutions to the practice problems) M 10/16 Solutions
17. Finding the Eigenvalues and Eigenvectors of a Matrix Problem Set 17 Bretscher 7.2 and 7.3, but skip Theorem 7.3.6 and Example 6 in the 4th edition / Theorem 7.3.5 and Example 5 in the 5th edition; Complex Numbers (see the Handouts page for solutions to the practice problems) W 10/18 Solutions
18. Diagonalization Problem Set 18 Bretscher 7.4 through Example 5 and Bretscher 7.5 through Example 5 if you have the 4th edition, Bretscher 7.1 and Bretscher 7.5 through Example 5 if you have the 5th edition F 10/20 Solutions
19. Diagonalization, Continued Problem Set 19 Bretscher 7.5 M 10/23 Solutions
20. Orthogonal Matrices, Symmetric Matrices and the Spectral Theorem Problem Set 20 Orthogonal Matrices, Symmetric Matrices and the Spectral Theorem W 10/25 Solutions
21. Introduction to Continuous Dynamical Systems Problem Set 21 Modeling with Systems of Differential Equations and Bretscher 9.1 F 10/27 Solutions
22. Linear Continuous Dynamical Systems and the Matrix Exponential Problem Set 22 A Brief Introduction to the Matrix Exponential M 10/30 Solutions
23. Linear Continuous Dynamical Systems, Continued Problem Set 23 Bretscher 9.2 W 11/1 Solutions
24. Nonlinear Continuous Dynamical Systems Problem Set 24 Bretscher "9.4" and Linearization from Differential Equations by Blanchard, Devaney, and Hall F 11/3 Solutions
25. Introduction to Linear Differential Equations Problem Set 25
Reading for #5: Linear Spaces and our annotated copy of Bretscher §4.1
the worksheet "Introduction to Linear Differential Equations", Modeling Springs with Differential Equations M 11/6 Solutions
26. Linear Spaces Problem Set 26 Linear Spaces and our annotated copy of Bretscher §4.1 W 11/8 Solutions
27. Linear Transformations Problem Set 27 Bretscher 4.2 through Example 2 only F 11/10 Solutions
28. Linear Differential Equations Problem Set 28 Bretscher 9.3 through Theorem 9.3.10 (you may skip the operator approach to solving linear diferential equations) or Linear Differential Equations You may turn this in M 11/20 with PS 31 (PS 30 will be due F 11/17) Solutions
29. More Practice with Linear Spaces and Linear Transformations No problem set!
30. Inner Product Spaces Problem Set 30 Bretscher 5.5 through Example 7 F 11/17 Solutions
31. Trigonometric Polynomials and Fourier Analysis Problem Set 31
Bretscher 5.5, starting right after Example 7 M 11/20 (remember to also turn in PS 28) Solutions
32. Fourier Series Problem Set 32
Fourier Series M 11/27 Solutions
33. Introduction to Partial Differential Equations Problem Set 33 The Heat Equation and the Wave Equation, §1 and 2 W 11/29 Solutions
34. The Heat Equation Problem Set 34 The Heat Equation and the Wave Equation, §3 F 12/1 Solutions
35. The Wave Equation Problem Set 35 The Heat Equation and the Wave Equation, §4 Due M 12/4 by 1 pm to your section's mailbox in the 2nd floor hallway near SC 221 Solutions

Note: Solutions to in-class worksheets can all be found on the worksheets page.