# Linear Algebra and Multivariable Calculus

## Overview

The pathways to advanced mathematics courses all begin with linear algebra and multivariable calculus, and the standard prerequisite for most linear algebra and multivariable calculus courses includes two semesters of calculus.

## Pre-requisites

Linear algebra and multivariable calculus can be taught using different approaches, so it is important to pay attention to course prerequisites. MATH 2210-MATH 2220 uses tools and techniques developed in linear algebra (MATH 2210, taken first) to develop multivariable and vector calculus (MATH 2220). MATH 1920-MATH 2930-MATH 2940 is designed for engineers and physical sciences students who will need some of the concepts from vector calculus early in their physics courses, and so the sequence begins with multivariable calculus (MATH 1920) and assumes no prior knowledge of linear algebra.

Course prerequisites are meant to be completed in their entirety before attempting the next course in the sequence. Students who attempt a course without adequate preparation often find themselves struggling at some point in the semester. This can be true even for students who have been highly successful in their math courses in the past. Taking two courses simultaneously when one is a prerequisite for the other is not advised.

## MATH 2210-MATH 2220

MATH 2210 is primarily a linear algebra course, but it also provides an introduction to linear ordinary differential equations. Some of the linear algebra in MATH 2210 is then used to develop multivariable and vector calculus in MATH 2220.

MATH 2210-MATH 2220 is taught at a higher theoretical level than MATH 1110-1120. For example, in MATH 2210 certain abstract concepts such as vector spaces are introduced, theorems are carefully stated, and many of these theorems are proved. MATH 2210 does not provide adequate preparation for MATH 2240.

MATH 2210-MATH 2220 is good preparation to begin a math major. Half of Cornell math majors take this sequence.

## MATH 2230-MATH 2240

MATH 2230-MATH 2240 (Theoretical Linear Algebra and Calculus) is “integrated” in the sense that both linear algebra and multivariable calculus are taught in each semester, rather than covered in separate semesters (as in MATH 2210-MATH 2220). The sequence is taught at a higher level of mathematical sophistication than MATH 2210-2220, particularly the portion dealing with vector calculus.

Only 17% of Cornell math majors take this sequence. The most successful students in MATH 2230-MATH 2240 begin with extensive mathematical experience and interest. Many have seen some vector calculus and linear algebra, and most are comfortable writing proofs. Those who find MATH 2230 too challenging or time-consuming may drop down to MATH 2210 at the beginning of the term or may continue with MATH 2220 instead of MATH 2240. Students who are considering MATH 2230 but are unsure if it is the right choice should pre-enroll in MATH 2210 and talk to an advisor after arriving at Cornell, then may transfer into MATH 2230 if advised to do so.

## Engineering Mathematics

The defining characteristic of the engineering mathematics sequence, MATH 1910-MATH 1920-MATH 2930-MATH 2940, is its rapid and utilitarian coverage of a wide variety of mathematical topics important in engineering and the physical sciences. Although the sequence covers more topics than other programs, the courses in the sequence cover those topics in somewhat lesser depth and with less attention to theoretical background, particularly when compared to MATH 2210-MATH 2220 and MATH 2230-MATH 2240. MATH 1920 covers multivariable calculus, MATH 2930 is a course in ordinary and partial differential equations, and MATH 2940 is a linear algebra course. MATH 1920 is the prerequisite for both MATH 2930 and MATH 2940, which are independent and can be taken in either order.

MATH 1920-MATH 2940 is a perfectly viable route to the math major. As these classes delve less into theory than MATH 2210-MATH 2220 and MATH 2230-MATH 2240, students who take them are strongly recommended to consult a math advisor before taking their first 3000-/4000-level math class. In particular, the department suggests MATH 3040 as a first upper-level course.

## MATH 2130 and MATH 2310

MATH 2130 (spring only, not offered 2023-2024) is a course with many different topics, including differential equations and multivariable calculus. MATH 2310 (fall only, not offered 2023-2024) is an applied linear algebra course that is less theoretical than MATH 2210. The only prerequisite for MATH 2310 is MATH 1110, but it is also a good option for students who have taken MATH 2130 and want to learn some linear algebra. Both courses are intended for students who do not expect to take more advanced math courses.

## Transfer Credit and Placement

Mathematically advanced high school students who have taken a linear algebra or multivariable calculus course will not receive transfer credit if the course was taught in a high school to high school students (even if the college provides a transcript) or if the course is used to satisfy high school graduation requirements.  Students may wish to take these courses again at Cornell, as the material is typically covered with greater depth and rigor than in advanced high-school courses. Students who have completed a rigorous course in multivariable calculus may take the Engineering Math Advanced Standing Exam to receive credit and placement out of MATH 1920. There is no placement test for linear algebra, and it should be noted that 4000-level linear algebra courses are generally not regarded as meeting the prerequisite for the math major or minor.  Finally, students who have background in both linear algebra and multivariable calculus and who would like an intensive, rigorous, abstract treatment of the material should consider our sequence MATH 2230-MATH 2240.  This year-long sequence is proof-based and includes material on topics such as manifolds and differential forms which are often reserved for upper division math courses.

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