Math 416, Abstract Linear Algebra (Spring 2020)

### Instructor: Pierre Albin

### Office: Illini Hall 237

### Email: palbin [at] illinois .edu

### Lectures:

(D13) MWF 11-11:50 Altgeld 341

(E13) MWF 13-13:50 Altgeld 241

### Office Hours:

Monday 12-12:50 and
Tuesday 14-14:50

### CORONAVIRUS Updates: Changes in how the class will work will
be posted on Piazza

### Assignments:
There will be homework each week due at the beginning of class on
Wednesday. You are allowed (and encouraged) to work
with other students while trying to understand the homework problems.
However, the homework that you hand in should be your work alone.
Late homework will not be accepted, but the two lowest scores will be dropped.

### Exams:
There will be four midterms (held outside of class) and a final.
We will drop the lowest midterm score.

The midterms will be held at **314 Altgeld Hall, 19:00-20:30,** on the
following dates:

** Feb 28, Mar 27, April 17, May 1.**
The final exam will be held at TBA.

### Holidays:
Classes begin on January 21 and end on May 6.
There will be no classes on:

**Exam replacement**, January 29- 31 and February 24-26,

**Spring break**, March 16 - March 20

### Grading percentages:

Problem sets (20%)

Midterms (40%)

Final exam (40%)

### Description:

Linear systems of equations show up in all areas of science and
their study has a long history.
A Babylonian clay tablet from around 300 BCE says roughly
There are two fields whose total area is 1800 square yards. One
produces grain at the rate of 2/3 of a bushel per square yard while
the other produces grain at the rate of 1/2 a bushel per square yard.
If the total yield is 1100 bushels, what is the size of each field?

One of the earliest surviving mathematical texts from China, "Jiuzhang
Suanshu" or "Nine Chapters on the Mathematical Art"
(written before 200 BCE) starts
its eighth chapter with the following problem:
A combination of 3 bundles of high-quality grain, 2 bundles of
medium-quality grain, and 1 bundle of low-quality grain will yield
39 barrels of flour. If we combine 2 bundles of high-quality grain,
3 bundle of medium-quality grain, and 1 bundle of low quality grain
we obtain 34 barrels of flour. Finally, combining 1 bundle of
high-quality grain, 2 bundles of medium-quality grain and 3 bundle
of low-quality grain we obtain 26 barrels of flour. How much flour
can be obtained from one bundle of each type of grain?

In modern algebraic notation we might write this as
3h+2m+1ℓ = 39

2h+3m+1ℓ = 34

1h+2m+3ℓ = 26

The method we use today to solve these systems, called Gaussian
elimination in honor of Carl Friedrich Gauss (1777-1855), is the same
method that was used in ancient China!

Eventually it was realized that it is better to think of a system of
equations as a single equation for a vector:
\begin{pmatrix} 3 & 2 & 1 \\ 2 & 3 & 1\\ 1 & 2 & 3 \end{pmatrix}
\begin{pmatrix} h \\ m \\ \ell \end{pmatrix}
=
\begin{pmatrix} 39 \\34 \\ 26 \end{pmatrix}.

Then it turned out that matrices were better understood not just as
coefficients of systems of equations but as maps that transform one
vector into another. This led to considering linear transformations
abstractly and so to the subject known as linear algebra.

Math 416 is a rigorous, abstract treatment of linear algebra.
Topics to be covered include vector spaces, linear transformations,
eigenvalues and eigenvectors, diagonalizability, and inner product
spaces. The course concludes with a brief introduction to the theory
of canonical forms for matrices and linear transformations.