Note that Apostol writes $V_3$ for what we have called $\R^3$ in class. The standard notation $\R^3$ was introduced after Apostol wrote his book. Similarly, he writes $V_n$ for what now is called $\R^n$.
Homework: §1.5. Problems 1–27 ask you to verify that some space is a vectorspace. Do problems 3, 5, 11, 12, 17, 22, 23. In each case make sure you describe the set $V$ which contains the vectors, and that you can describe how vector addition and multiplication with numbers is defined.
Also do Problem 31.
The “strange example” described in class is problem 29.
Homework: §1.10. Problems 4, 5, 6, 7; 11, 12, 14, 16, 17, 19. What is the difference between problems 19 and 20?
Homework: §1.10. Problems 22, 23, 24. Note that Apostol writes $L(S)$ for what we have been calling the span of the set $S$
Homework: §1.13. Problems 1, 3, 4, 5, 8, 10, 12.
§2.8 (page 42) 23,25, 28ab
§2.12 (page 50) 1, 2, 3, 4, 5, 11, 12, 14.
§3.6–1ac, 2a, 3a, 4abd, 9, 10.
§3.11–1, 7a.
§3.17–1ab, 5a, 6a
§3.17–1c: You are asked to find the cofactor matrix of a $4\times4$ matrix. The definition requires you to compute sixteen $3\times3$ determinants. Instead of doing this, compute the determinant, and the inverse of the matrix using the computational scheme from page 66 (§2.19). If you know the inverse and the determinant, how do you get the cofactor matrix?
Here is the list of topics and problems in preparation for Thursday’s midterm
Here is an old second midterm from 2004.
Differentiation. Directional and partial derivatives. Higher partial derivatives. (8.6, 8.7, 8.8).
§ 4.4, page 101: problems 1,2,3,4,11.
§ 4.8, page 107: problems 2,3, 6, (12 was done in class), 14
§ 4.10, page 113: problems 4, 7, 8.
Review problems on matrices and eigenvalues from math 519
§8.9, page 255: problems 1, 2a, 4—9, 10, 11, 14 (note: $D_1f$ is Apostol’s notation for the derivative with respect to the first argument; in these problems $D_1f = \frac{\partial f}{\partial x}$ ).
§8.14, page 262: problems 1, 2, 6, 7bc, 8.