###### What is Linear Algebra?

Linear Algebra is the study of linear maps between vector spaces and the matrix algebra necessary to apply the theory to concrete problems. Its roots are quite modest: solving systems of linear equations. Linear algebra, however, has developed substantially and permeates most of modern mathematics and much of modern science and engineering. In this class, we will cover many topics of linear algebra, including Gaussian elimination, matrix algebra, $LU$ factorization, vector spaces, subspaces, dimension, norms and inner products, positive definite matrices, minimization, nearest point and least squares, data fitting, orthogonality, Gram-Schmidt method, the Fast Fourier Transform, and the revolutionary page rank algorithm.

dataBar EmptyHomework assignments will be posted here. The homework will not be collected, and instead there will be weekly quizzes on the material. Quizzes and solutions will be posted here after the quiz is taken.

**Quiz solutions:** Quiz 1, Quiz 2,
Quiz 3, Quiz 4 was take-home, Quiz 5,
Quiz 6, Quiz 7, Quiz 8,
Quiz 9

**Midterms:** Midterm 1 Solutions,
Review 2,
Midterm 2 Solutions,
Final Review

**Assignment 1:** __Section 1.2__ # 1-3, 6, 7begh, 12 (optional challenge question), 18, 20

**Assignment 2:** __Section 1.3__ # 1 becf, 3cdfg, 13 (challenge), 14, 16, 21a, 23, 32 cdef
__Section 1.4__ # 19 bdef, 25 (challenge)

**Assignment 3:** __Section 1.5__ # 3, 4 , 5, 9, 11 (challenge), 15, 18, 20
(Note: #4 is related to the mistake I made in class!)
__Section 1.8__ # 1ace, 2cdfg, 7, 9, 10, 12 (related to last class example), 23, 24

**Assignment 4:** __Section 1.9__ # 1, 2, 3, 5, 6, 12 (Hint: use Thm 1.50. Remember that multiplying by an
elementary matrix on the left is the same as performing a row operation)
__Section 2.1__ # 1, 2, 8, 13. __Section 2.2__ # 1, 7, 8, 10

**Assignment 5:** __Section 2.3__ # 3, 4, 5, 7 (Note: a matrix $S$ is symmetric if
it matches its transpose; $S^T = S$), 8, 14a, 16, 17, 19 (Note: (a)
is related to main idea in the proof of Lemma 2.30), 21, 30
__Section 2.4__ # 1, 2, 4, 14 (Hint: use Lemma 2.24),
15, 21, 25ab (Hint: Use Lemma 2.30)
__Section 2.5__ # 1, 2, 3, 5, 12, 29 (Hint: Form the corresponding
matrices and check the compatibility conditions after row reducing), 36, 38, 39.

**Assignment 6:** __Section 3.1__ # 2, 7, 9, 10, 11, 19, 22 (Hint:
the formula *does* define an inner product on $\mathcal{P}^{(n)}$. Why?),
25, read 29 and then answer 30
__Section 3.2__ # 1ab, 3, 8, 18, 25, 19 (do 25 first), 27, 39, 40
__Section 3.3__ # 4, 11, 13, 14 (challenge)

**Assignment 7:** __Section 3.4__ # 1 (Hint: Use the result of
the next section, Thm 3.37), 7, 10, 11. __Section 3.5__ # 1, 3a
__Section 4.2__ # 1, 3abfg, 5, 6, (challenge: do 6 with general $n$, i.e., solve
problem 6 for ** all** $n$.)

**Assignment 8:** __Section 4.3__ # 1, 2, 3, 7 (Note: as is listed on the top of
these exercises, "distance" refers to the standard $L^2$ distance unless otherwise indicated.)

**Assignment 9:** __Section 4.3__ # 14 bd, 15. __Problem__: Find * all* least squares solutions to the system
$Ax = b$ where

- $$A = \begin{pmatrix} 2 & -4 & 1 & -1\\ 1 & -2 & -2 & 0\\ 3 & -6 & -1 & 1 \end{pmatrix} \quad \mbox{and} \quad b = \begin{pmatrix} 1\\0\\1\end{pmatrix}.$$
- $$A = \begin{pmatrix} 2 & 3 & 1\\ 4 & -2 & 10\\ 1 & 5 & -3\\ 2 & 0 & 4\\ \end{pmatrix} \quad \mbox{and} \quad b = \begin{pmatrix} 2\\-1\\1\\3 \end{pmatrix}.$$

**Assignment 10:** __Section 4.4__ # 1, 2, 3 (but, as 2008 shows, don't buy a house based on this model!)
4-6 (Hint: For problems 4-6, look at example 4.11),
12 bd(just construct the inerpolating polynomials: don't pay attention to the
"Lagrange form"), 13bd, 15, 16 (Hint: objects fall according to
to a quadratic polynomial), 31, 32, 33, 35.

**Assignment 11:** __Section 5.1__ # 1abdf, 3abdf, 6, 10, 14, 21, 22, 23
__Section 5.2__ # 1, 4, 7.
__Complex arithmetic__ Do some problems at
this link.
(Hint: the quadratic formula holds for complex equations; see the lecture notes
for division with complex numbers.)

**Assignment 12:** # 1 Compute the sampled exponential bases $\omega_0,\omega_1,\ldots, \omega_{n-1}$ for
$n = 3, 4, 6, 8, 12$. # 2 Verify when $n=4$ that the sampled exponentials $\omega_0,\omega_1,\omega_2,
\omega_3$ form an orthonormal basis under the averaged dot product. __Section 5.7__ # 1, 2, 3 (for $n=4,8$) (
Note: The sign function $\mathrm{sign}(a)$ is 1 for $a> 0$, -1 for $a<0$,
0 for $a=0$.)

**Assignment 13:** This take-home quiz

**Week 1, 9/3-9/5:** 1.1-1.3; Systems of linear equations, matrices, Gaussian Elimination.
Lecture 1, Lecture 2

**Week 2, 9/8-9/12:** 1.3-1.5; $LU$ factorization, permuted $LU$ factorization, inverse matrices.
Lecture 3, Lecture 4, Lecture 5

**Week 3, 9/15-9/19:** 1.8-1.9; General linear systems, homogeneous equations, determinants.
Lecture 6, Lecture 7, Lecture 8

**Week 4, 9/22-9/26:** 2.1-2.3; Vector spaces, subspaces, span. Lecture 9,
Lecture 10, Lecture 11

**Week 5, 9/29-10/3:** Workshops on Proofs | 2.3; Linear Independence.
Lecture 12 / Workshop 1, Lecture 13 / Workshop 2,
Lecture 14

**Week 6, 10/6-10/10:** Bases, dimension, matrix subspaces. Lecture 15,
Lecture 16,
Lecture 17

**Week 7, 10/13-10/17:** Application—Solving PDE numerically, Review, Midterm 1 on 10/17.

**Week 8, 10/20-10/24:** 3.1-3.3; Inner products, related inequalities, norms.
Lecture 19,
Lecture 20,
Lecture 21

**Week 9, 10/27-10/31:** 3.4, Thm 3.37, 4.2, 4.3; Positive definite matrices, minimization of quadratic functions, nearest point.
Lecture 22,
Lecture 23,
Lecture 24

**Week 10, 11/3-11/7:** 4.3; Nearest point, least squares.
Lecture 25,
Lecture 26

**Week 11, 11/10-11/14:** Data fitting, interpolation, orthogonality.
Lecture 27,
Lecture 28,
Lecture 29

**Week 12, 11/17-11/21:** Gram-Schmidt process, complex vector space, review.
Lecture 30,
Lecture 31

**Week 13, 11/24-11/28:** Midterm 2 on 11/24. No class 11/26.

**Week 14, 12/1-12/5:** Discrete Fourier analysis, the fast Fourier transform.
Lecture 32,
Lecture 33,
Lecture 34,
Fourier Transform Examples,
Fourier transform in action

**Week 15, 12/8-12/12:** Google and page rank, review.
Lecture 35

Final Exam 12/16: 1:30pm-3:30pm in room our regular classroom.

**Quizzes:**25%

**Midterm 1:**25%

**Midterm 2:**25%

**Final Exam:**25%

This table shows the grading weights for this class. Notice that the homework is not formally part of your grade, but to succeed in this class, you will need to master the theory and techniques. The best way to do this is through doing the homework, studying together, and coming to office hours with questions.

The class will be graded relative to a curve, with each exam curved individually and the quizzes curved as a whole. The lowest 2 quiz grades will be dropped to accommodate an absence, bad day, etc... However, if you need to miss a class due to a religious observance, let me know in advance and we can schedule a make-up quiz.

**Curve for the first exam:** Take your score out of 50, divide by 10. This is your score on the 4.0 scale.
For example, if you received a 32/50, this translates to a 32/10 = 3.2 on the 4.0 scale, or a high B. If you received a
25/50, this translates to a 25/10 = 2.5, or somewhere between a C+ and a B-.

**Instructor:**Stephen Lewis

**Website:**stephen-lewis.net

**Office Hours:**W 12:30-2:30, Th 2-4, or by appointment

**Office:**Vincent Hall 221 (Map)

**Email:**iam@stephen-lewis.net

Syllabus

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