Math 21b - In-Class Worksheets

Course head: Janet Chen
Course preceptor: Yu-Wen Hsu (

Not all sections of Math 21b will use worksheets, and worksheets will vary from section to section. Here, we'll post one sets of worksheets, with solutions. Most of the worksheets are (intentionally!) much too long to be completed in class; this way, you always have a source of additional practice problems. We encourage you to use these extra problems whenever you feel like you want extra practice with a topic.

1Introduction to Linear Systemssolutions
2Gauss-Jordan Eliminationsolutions
3Introduction to Linear Transformationssolutions
4How much data do you need to determine a linear transformation?solutions
5More Examples of Linear Transformationssolutions
6More on Bases of \(\mathbb{R}^n\), Matrix Productssolutions
7Matrix Inversessolutions
9Image and Kernel of a Linear Transformation, Introduction to Linear Independencesolutions
10Subspaces of \(\mathbb{R}^n\), Bases and Linear Independencesolutions
11Dimension and the Rank-Nullity Theoremsolutions
12Orthogonal Projections and Orthonormal Basessolutions
14The Gram-Schmidt Process, The Transpose of a Matrixsolutions
15Least Squares and Data Fittingsolutions
16Introduction to Discrete Dynamical Systems and Eigenanalysissolutions
17Finding the Eigenvalues and Eigenvectors of a Matrixsolutions
19Diagonalization, Continuedsolutions
20Orthogonal Matrices, Symmetric Matrices and the Spectral Theoremsolutions
21Introduction to Continuous Dynamical Systemssolutions
22Linear Continuous Dynamical Systems and the Matrix Exponentialsolutions
23Linear Continuous Dynamical Systems, Continuedsolutions
24Nonlinear Continuous Dynamical Systemssolutions
25Introduction to Linear Differential Equationssolutions
26Linear Spacessolutions
27Linear Transformationssolutions
28Linear Differential Equationssolutions
29More Practice with Linear Spaces and Linear Transformations

Also, here are Chi-Yun's worksheet and solutions.

30Inner Product Spacessolutions
31Trigonometric Polynomials and Fourier Analysis

Here's what the Fourier approximations for #6 look like (showing \(\text{proj}_{T_n} f\) for different values of n)

32Fourier Seriessolutions