MATH 308 M: MATRIX ALGEBRA

Resources

Final

Here's the final: without solutions; with solutions. Here are some statistics.

Here are two practice finals with solutions. We will go over part of the first in lecture.

• Practice final A; solutions. We did not explicitly cover diagonalization, so skip 1(e), 2(d) (you can do 6).
• Practice final B; solutions. We did not explicitly cover diagonalization, so skip 1(j). Replace the directions for 5 with: "find the value of k for which the sum of the dimensions of the eigenspaces of A is 3 and find bases for these eigenspaces".

Midterm 2

Here's Midterm 2: without solutions; with solutions. Here are some statistics.

Here are two practice midterms with solutions. We will go over the first in lecture.

• Practice midterm A; solutions. This exam includes questions on determinants, which you should skip.
• Practice midterm B; solutions. In this exam, N(A) denotes the null space of the matrix A and R(A) denotes the range of the linear transformation associated to A (not the row space of A).

Midterm 1

Here's Midterm 1: without solutions; with solutions. Here are some statistics.

Here are two practice midterms with solutions. We will go over the first in lecture.

• Practice midterm A; solutions
• Practice midterm B; solutions

Alternatives to the Book

While we will cover most of Holt in order, the underlying mathematics is quite old and there are many, many alternative expositions. A few:

• S. Paul Smith has extensive and reasonably well-edited notes which are written at a somewhat more advanced level than Holt. See the "sermon" at the end as well.
• Natalie Naehrig has some perhaps friendlier notes, which are not quite as polished.
• Alex Young has turned the course into a series of "Paul's online math notes"-style pages.
• Khan Academy has extensive material at this level, like this.

Proofs

The course is "Matrix algebra with applications", though perhaps "Applications with matrix algebra" is more accurate--the emphasis is on computational rather than theoretical aspects of elementary linear algebra. Nonetheless, proofs will be presented in lecture and you will be asked to give some proofs on exams.

To successfully construct proofs may require a change in your perspective. Being "mostly right" is fine in much of life, but not in math, and especially not when writing proofs. Misunderstanding a definition or forgetting to check an assumption on a theorem can turn your argument into complete nonsense. Here are some examples from students of the difference between "mostly right" definitions and fully rigorous ones.

Even after getting definitions and theorems down perfectly, constructing proofs still requires creativity and skill. The proofs you will be asked to give in this course will all be quite short, likely a few lines long, and they will largely be direct consequences of the definitions and theorems. Hence the proofs in this class will emphasize understanding of course material rather than developing advanced proof-making skills. Some time in lecture will be devoted to writing your own proofs and critiquing each others' proofs.

Proofs

See the example proofs document for (shockingly) some example proofs. See homework for information on proof-based homework.

Inverses

Here is a summary of the course's material on inverses.

Diagonalization

Here is a summary of the course's optional material on diagonalization, which will not be on the final.