The introductory sequence of courses for the mathematics major involves a course in Analysis and a course in Linear Algebra. There are two paths which students can take to cover these major prerequisites. Both will prepare students well for the mathematics major and for upper-level electives which build upon these foundational areas in mathematics.
(MATH 222 or MATH 225) & MATH 250: The first track is a two-course sequence involving either MATH 222 (Linear Algebra with Applications) or MATH 225 (Linear Algebra and Matrix Theory) and MATH 250 (Vector Analysis). MATH 222 or MATH 225 covers the Linear Algebra aspect of the introductory sequence. Both courses examine the notions of vector spaces and linear transformations before exploring topics such as matrices and Gaussian elimination, solving systems of linear equations, the determinant, null space, rank, diagonalization, eigenvectors, eigenvalues, and eigenspaces. The largest distinction between MATH 222 and MATH 225 is the use of proofs; MATH 225 is an entirely proof-based course, where the majority of work will consist of constructing logical arguments to explain why something is true rather than just finding an answer. In MATH 225, the answers to problem sets will be proofs and paragraphs, not only computational results. For students who are planning on continuing in mathematics, the construction of proofs and the requisite skill of logical thinking will be invaluable. If you elect to choose the MATH 225 track, Linear Algebra will be your first exposure to proof-based mathematics. If not, your first introduction to proofs will be in MATH 250; this course is a rigourous introduction to vector analysis and the calculus of functions of several variables. It will investigate the derivative as a linear mapping, the inverse and implicit function theorems, transformation of multiple integrals, line and surface integrals of vector fields, curl and divergence, differential forms, and the theorems of Green and Gauss along with the generalized Stokes’ theorem
MATH 222 is taught with Strang’s Linear Algebra and its Applications. You can preview the syllabus for the Spring 2018 course here. MATH 225 is taught with Linear Algebra by Friedberg, Insel, and Spence. You can preview the syllabus for the Spring 2018 course here. MATH 250 is taught with Hubbard and Hubbard’s Vector Calculus, Linear Algebra, and Differential Forms: A unified approach.. A syllabus from the Fall 2017 course can be found here.
MATH 230/231: The second track open to students is a two-term combined course sequence which covers vector analysis and linear algebra in a single class. These courses cover the foundations of real analysis, topology, and number theory before using these tools to lay down a rigorous foundation for multivariate analysis and linear algebra. Unlike MATH 222, this course sequence is entirely proof-based; as with MATH 225, students will not only use advanced computational techniques but will learn to construct logical arguments for the topics covered in class. The analysis portion of MATH 230 will cover all topics covered in MATH 120, as well as some additional topics including (but certainly not limited to) the Banach and Brouwer Fixed Point Theorems, Cauchy sequences, Lipschitz functions, the generalized Stokes’ Theorem, and manifolds. The linear algebra portion of the class covers all topics covered in MATH 225, along with connections between linear algebra and analysis, such as tangent spaces. Along with providing a solid foundation for analysis and linear algebra, MATH 230/231 often includes lessons chosen by each professor from their area of expertise. Some professors elect to include topics in discrete math and algebra.
MATH 230/231 is taught with Hubbard and Hubbard’s Vector Calculus, Linear Algebra, and Differential Forms: A unified approach. You can preview the syllabus for the 2017/2018 school year here.
Which track should I choose? Both sets of courses will give students adequate preparation to succeed in the department. Students electing to take either MATH 225 or MATH 230/231 will get an early exposure to proof-based mathematics, which is essential for continued study and research in the discipline. MATH 230/231 also requires a larger amount of work than MATH 250 and MATH 222/225 combined do; this can often mean longer, more intensive problem sets each week which enable the class to move at a faster pace and cover more topics in-depth than would otherwise be possible. If you are taking several classes with a high workload, take this into consideration. Additionally, MATH 230/231 is typically only offered at a single time each semester, while MATH 120 has multiple sections affording students much more flexibility than is possible with MATH 230/231; if you have a tight schedule, it may be difficult to have the free time-slot required to take MATH 230/231.
MATH 230/231 is like learning how to disassemble a car and re-build it out of coconuts. You’ll learn a lot about how to drive a car, but you’ll also learn a lot about how cars operate and what all the nitty-gritty details of how they work are.
— A MATH 230/231 Instructor
What is the work like? Both sequences of courses — indeed, much of the mathematics major — will involve a good deal of work. Often times, the work you do may be different from what high school math courses are like. Memorization of formulae and learning where and when to apply different techniques aren’t really the focus of mathematics. Rather, you will be learning how to think and reason about problems, some of which don’t have answers. This skill is at the core of mathematics research, and at the core of the classes you will take for your major. Regardless of which sequence you choose, you can expect that you’ll spend many hours trying to piece together proofs with statements which don’t seem related. Sometimes, theorems will seem trivially obvious, but their proofs will be frustratingly elusive. And that’s okay! Math isn’t easy, but it is rewarding, and it is a skill like any other. The only way to become better is to practice, and the classes you take at Yale will certainly give you plenty of that!
As an example of what might show up in your proof based analysis and linear algebra courses, we’ve collected some problems of the type which might be given in MATH 225, 230, and 250.
- Prove or disprove that there are infinitely many prime numbers.
- Prove that for all $x \not= 0$, if $xz = yz$, then $y = z$.
- Prove that the composition of continuous functions is continuous.
- Prove or disprove that $\text{GL}_n(\mathbb{R})$ is closed under transposition.
- Prove or disprove that if and only if $A$ is an invertible square matrix, then the left and right inverses must be equal to one another.
- Let $S$ be the set of $2\times2$ real matrices $$\displaystyle M = \begin{bmatrix} a & b \\ c & d \end{bmatrix} $$ where the entries $a,b,c,d$ (in that order) form an arithmetic progression. Find all matrices $M \in S$ for which there is an integer $k > 1$ such that $M^k \in S$.
- Prove the Bolzano-Weierstrass Theorem.
- Prove or disprove that the closure of an open discrete set cannot be discrete.
- Find an injection from $\mathbb{R}^n \to \mathbb{R}$, or prove that one cannot exist.
- Prove or disprove that a closed and bounded subset of $\mathbb{R}^n$ with the usual topology must be compact.
- Prove that there can be no bijection between a set and its power set.
- For each real $x$, let \[ f(x) = \sum_{n \in S_x} \frac{1}{2^n}, \] where $S_x$ is the set of all positive integers $n$ such that $\lfloor nx \rfloor$ is even. What is the largest real number $L$ for which $f(x) \geq L$ for all $x \in [0,1)$?
- Prove that the Cartesian product of countable sets must be countable.
- Is there a projective plane of order $12$?
Some of these problems are from the Putnam exam, a test given annually to college students with a median score of zero. Putnam exam problems sometimes work there way into our problem sets, and many problems are quite similar to those on the Putnam. You also might notice that a few of these problems are open problems, for which there is no known answer! Don’t worry, these are given as bonus questions, along with other, more solvable questions.