# Math 375 — Multi-Variable Calculus and Linear Algebra

## Lecture and Homework Schedule

### September

R7 Linear spaces. (1.2-1.4)

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.

T12 Subspaces. Linear independence. (1.6,1.7)

Homework: §1.10. Problems 4, 5, 6, 7; 11, 12, 14, 16, 17, 19. What is the difference between problems 19 and 20?

R14, T19, R21 Bases and dimension. Euclidean case. (1.8,1.9,1.11)

Homework: §1.10. Problems 22, 23, 24. Note that Apostol writes $L(S)$ for what we have been calling the span of the set $S$

T26, R28 Inner products. Orthogonality. Cauchy–Schwartz inequality. Gram–Schmidt (1.11,1.12)

Homework: §1.13. Problems 1, 3, 4, 5, 8, 10, 12.

### October

T3 First midterm (location to be announced)
R5 Nearest vector in a linear subspace; Fourier expansions.
Homework: §1.17. Problems 1, 2b.
Problems 3, 4. In the end these problems involve computing a bunch of integrals, but before you compute them explain why you have to compute them and what the results mean.
Problems 6, 8.
After tuesday: §2.4, problems 1—5, 7, 8, 10, 18, 19, 22.
T10 Linear transformations. (2.1, 2.2) Inverse transformation. (2.6,2.7) Matrix representation of linear transformation. (2.10,2.13,2.14)
R12–R26 Multiplication of matrices. Inverses. (2.15,2.19)
Homework: §2.4: 24, 25 (in 25 assume that $V$ is the space of polynomials instead of the space that Apostol specifies.)

§2.8 (page 42) 23,25, 28ab

§2.12 (page 50) 1, 2, 3, 4, 5, 11, 12, 14.

T31 Axioms for determinant. Uniqueness. Product formula. Determinant of the inverse. (3.3, 3.5, 3.7, 3.8)
Homework: (from chapter 3)

§3.6–1ac, 2a, 3a, 4abd, 9, 10.

§3.11–1, 7a.

### November

R2 Minors and cofactors. (3.12,3.14)
Homework:

§3.17–1ab, 5a, 6a

T7 Cramer's Rule. (3.15,3.16)
R9 Eigenvalues and eigenvectors, similar matrices. (4.2,4.6,4.9)
T14 Eigenvalues and eigenvectors, similar matrices. (4.2,4.6,4.9)
Homework:

§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

R16 Second midterm (location: in class).
T21 Eigenvalues and eigenvectors, trace and determinant.
R23 Thanksgiving
T28 Functions between Euclidean spaces. Quick description of Open sets, Limits, and Continuity. (8.1,8.2,8.4)

Differentiation. Directional and partial derivatives. Higher partial derivatives. (8.6, 8.7, 8.8).

Homework:

§ 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.

R30 Higher partial derivatives. (8.6, 8.7, 8.8).

### December

§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 $D_1f = \frac{\partial f}{\partial x}$ ).
R7&T12 Sufficient condition for differentiability (8.13); The Frechet derivative of $f:\R^n\to\R^m$, and the Jacobian matrix (8.18); Differentiability implies continuity (8.12 and 8.19); Chain Rule (8.15 and 8.20)