# 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

Here is an old second midterm from 2004.

**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

**T5**
Total derivative. Gradient. (8.11,8.12)

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.

**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)