Math 21b - Homework

Course head: Janet Chen
Course preceptor: Yu-Wen Hsu (
Here is the homework policy in a nutshell:

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
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
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
19. Diagonalization, Continued Problem Set 19 Bretscher 7.5 M 10/23
20. Orthogonal Matrices, Symmetric Matrices and the Spectral Theorem Problem Set 20 Orthogonal Matrices, Symmetric Matrices and the Spectral Theorem W 10/25

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