Harvard MATH 25: Theoretical Linear Algebra and Real Analysis
Math 25 (25A in fall, 25B in spring) is Harvard's honors first-year sequence in proof-based linear algebra and real analysis, the rigorous track below the legendary Math 55. It's the standard landing spot for strong students who want real mathematics at a survivable pace.
Fennie is independent and not affiliated with Harvard University. This is an unofficial study guide.
What makes it hard
It's most students' first fully proof-based course, and psets routinely run 10-20 hours while the proof-writing muscles develop. The pace is humane only relative to Math 55: epsilon-delta arguments, abstract vector spaces, and compactness still arrive faster than intuition forms.
What you'll cover
- • Proof techniques and rigor
- • Vector spaces and linear transformations
- • Eigenvalues and inner product spaces
- • Construction of the real numbers
- • Sequences, series, and continuity
- • Metric spaces and compactness
The MATH 25 study guide
How to study for Harvard MATH 25, step by step.
- 1
Drill definitions until they generate examples
Every Math 25 proof is assembled from definitions used precisely. After each lecture, restate the new definitions from memory and construct an example and a non-example for each. The non-examples teach more.
- 2
Write proofs daily, in small doses
Proof-writing is a motor skill that compounds with frequency. One carefully written proof per day, checked against feedback, builds the fluency psets demand faster than weekend marathons.
- 3
Start each pset the day it posts
The hard problems fall to spaced attempts: read everything immediately, let the stuck ones marinate overnight, and return across several days. Discussion with classmates is standard; write-ups are solo.
- 4
Reconstruct lecture proofs from memory
Closing the notes and reproving the week's main theorems is the highest-yield exam prep in the course, because pset problems are variations on exactly those techniques.
Today's MATH 25 plan
What a Fennie Daily Plan looks like for MATH 25. Yours is built from your syllabus and adapts every day to your deadlines and progress.
- Review: Eigenvalues and inner product spacesReview · 25 min
Work back through the MATH 25 material on eigenvalues and inner product spaces. In the app, Fennie builds this from your own notes and syllabus.
- Practice: Construction of the real numbersPractice · 20 min
Targeted problems on construction of the real numbers, the kind MATH 25 actually tests.
- Quick quiz: Sequences, series, and continuityQuiz · 10 min
Five generated questions to expose weak spots on sequences, series, and continuity before the exam does.
- Preview: Metric spaces and compactnessPreview · 10 min
A first pass over metric spaces and compactness so the next session starts from familiar ground.
First plan free, no card required. Fennie is independent and unaffiliated with your school.
FAQ
Is Math 25 hard?
Yes. It's fully proof-based from day one and psets commonly take 10-20 hours. It's deliberately gentler than Math 55 while covering rigorous linear algebra and analysis.
Math 25 or Math 55?
Math 55 assumes substantial prior proof experience and runs at extraordinary pace; Math 25 builds proof skills while covering rigorous material. Dropping from 55 to 25 early in the fall is common and unstigmatized.
What does Math 25 prepare you for?
It's a launchpad into upper-level math (abstract algebra, topology, and further analysis) and a strong foundation for theoretical CS, statistics, and physics tracks.
More Harvard courses
MATH 55: Studies in Algebra and Group Theory / Real and Complex Analysis
Math 55 (55A in fall, 55B in spring) is Harvard's legendary honors freshman math sequence, covering proof-based abstract algebra, group theory, and real and complex analysis at extraordinary speed and depth. It's widely described as the hardest undergraduate math class in the country.
MATH 1A: Introduction to Calculus
Math 1A is Harvard's introductory calculus course, covering limits, derivatives, integrals, and the Fundamental Theorem of Calculus, for students who didn't take or don't have credit for calculus before college. It leads into Math 1B and the rest of the math sequence.
MATH 21A: Multivariable Calculus
Math 21A covers multivariable calculus (vectors, partial derivatives, multiple integrals, and vector fields) and is the standard math course for Harvard students in sciences, economics, and pre-med tracks who arrive with single-variable calculus done.
MATH 21B: Linear Algebra and Differential Equations
Math 21B covers linear algebra (systems, matrices, eigenvalues, orthogonality) and applies it to differential equations and Fourier series. It's a workhorse requirement for applied math, economics, CS, and science concentrators.