In order to complete the major in mathematics you must take a course in linear algebra. At UW-Madison, we offer several versions of linear algebra. **Note that in all versions of the major and certificate, only one of the following courses may be used to fulfill any of the requirements.**

The purpose of this page is to describe the essential differences between the four introductory-level linear algebra courses.

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## MATH 320: Linear Algebra and Differential Equations

MATH 320 covers both some linear algebra and some differential equation theory. As such, students who complete this course can consider themselves as also having some of the content of MATH 319 (Techniques in Ordinary Differential Equations). The difference between this course and taking both MATH 319 and 340 is that one will be able to see how theory and applications unite in a meaningful way. This course also lends itself to the MATH 321-322 applied analysis sequence.

Students who have completed MATH 320 will need to complete one of MATH 421; the applied analysis sequence MATH 321-322; or MATH 467 before moving on to 500-level courses that require introductory proofs coursework.

In a regular fall or spring semester, MATH 320 is often taught with both general sections and an Accelerated Honors (!) section. There are some differences between the two:

#### General MATH 320

Differential equations and linear algebra are crucial for the analysis and control of many dynamical systems such as electrical networks, mechanical structures, chemical and biological networks. Those applications lead to often very large systems of coupled differential equations. The study of the equilibria, stability and control of such systems requires the concepts and tools of differential equations and linear algebra. MATH 320 is a first course in the study of differential equations and linear algebra.

#### MATH 320 Honors

Accelerated Honors at UW-Madison are Honors courses open to anyone and not restricted to students in the L&S Honors program. MATH 320 Honors is recommended for students in all applied areas including all areas of engineering, as well as the physical, chemical, and biological sciences. Differential equations and linear algebra are crucial to the study of dynamical systems such as electrical networks, mechanical structures, and chemical reaction networks. MATH 320 Honors has recently used Gilbert Strang’s *Differential Equations and Linear Algebra* textbook. MATH 320 Honors covers some material that is important for engineering and data science applications but may not be covered in the general (non-Honors) MATH 320 lectures such as, for example, the Dirac delta function and impulse response, the PLU and CR matrix decompositions, together with an introduction to the use of modern computational tools such as MATLAB (or Python).

In summary, MATH 320…

- Is useful for students interested in classical applications of mathematics (e.g., physics, engineering, continuous modeling, etc.);
- Covers material in MATH 319 and therefore
**credit for only one of MATH 319 or 320 can be applied to the math major or certificate;** - Is not by itself sufficient for taking advanced math courses;
- Is a good introduction to how theory and applications support each other;
- Is offered with an Accelerated Honors (!) version in some semesters. This version is suggested for potential math majors and those in the AMEP program.

Suggested further courses are…

- The applied analysis sequence MATH 321-322, which covers more mathematics useful for traditional applications;
- MATH 415, which includes both continuous and discrete models of changing systems;
- MATH 421 for an introduction to more formal mathematical arguments.

## MATH 340: Elementary Matrix and Linear Algebra

MATH 340 is a basic linear algebra course which focuses on vectors as ordered sets of real numbers and linear operators as matrices. In this course the focus is typically on computational aspects of the subject with some lighter treatment of the more theoretical points.

Students who complete this course and would also like exposure to differential equations should consider MATH 319.

Students who have completed MATH 340 will need to complete one of MATH 421; the applied analysis sequence MATH 321-322; or MATH 467 before moving on to 500-level courses that require introductory proofs coursework.

In summary, MATH 340…

- Is ideal for students who need functional knowledge of basic matrix algebra, and in particular those looking for applications featuring discrete mathematics (i.e., computer science and possibly statistics);
- Is not by itself sufficient for enrollment in advanced math courses.

Suggested further courses are…

- MATH 319 for those interested in the applied analysis sequence;
- MATH 421 for those interested in advanced undergraduate math courses above the 500-level.

## MATH 341: Linear Algebra

MATH 341 is a linear algebra course which is also meant to be an introduction to proofs and proof-writing. The linear algebra content of the course is more robust than any of the others listed on this page. Students who complete the course should be well-prepared to move on to any upper-level course, in particular MATH 521, 541, or 551.

It is the recommended linear algebra course for majors interested in moving to advanced undergraduate courses quickly.

**Due to the more intensive proof-writing nature of the course, students will probably find the course more demanding than MATH 340, and for this reason MATH 341 carries the Accelerated Honors (!) label.**

Students who complete this course and would also like exposure to differential equations should consider MATH 319.

In summary, MATH 341…

- Is accepted in both the major and certificate programs;
- Is a good introduction to proofs and proof-writing;
- Will give students access to advanced-level undergraduate math courses.

Suggested further courses are…

- MATH 421 for another exposure to formal mathematical arguments at the introductory-level;
- Any math course above the 500-level (possibly assuming other prerequisites).

## MATH 375: Topics in Multi-Variable Calculus and Linear Algebra

MATH 375 is an Accelerated Honors (!) course which features the role that linear algebra has in multivariable calculus.

It is assumed that students who complete this course will move on to complete the sequel course, MATH 376. (Students who complete MATH 375 and not MATH 376 are not considered to have completed the content of MATH 234! So by enrolling in MATH 375 in the fall, you should be prepared to enroll in MATH 376 in the spring, or MATH 234 in order to complete multivariate calculus.)

In summary, MATH 375:

**Is Honors-level;**- Enrollment is by permission only;
- Is not a course you should take if you have credit for one or more of MATH 234, 319, 320, 340, or 341;
- Is a good introduction to proofs and proof-writing.

More information on the MATH 375/376 sequence can be found on our Honors calculus page.