Math 416, Abstract Linear Algebra (Spring 2020)

(D13) MWF 11-11:50 Altgeld 341

(E13) MWF 13-13:50 Altgeld 241

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

The midterms will be held Feb 28, April 3, and April 17.

The final exam will be held on May 8.

Problem sets (20%)

Midterms (40%)

Final exam (40%)

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 = 39The 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!

2h+3m+1ℓ = 34

1h+2m+3ℓ = 26

Eventually it was realized that it is better to think of a system of equations as a single equation for a vector:

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.\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}.

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.