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.

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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.
5
Protect the proof hours with Fennie
Upload the Math 25 syllabus and Fennie's Daily Plan reserves consistent daily blocks for pset progress and proof reconstruction, with flashcards for the accumulating definitions generated from your actual notes. Free to start.
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How Fennie helps with MATH 25

Fennie's Daily Plans protect the consistent daily proof-writing blocks Math 25 actually requires, instead of letting psets collapse into weekend marathons. Chat through an epsilon-delta argument until it's load-bearing, and drill flashcards on the definitions that every proof quietly assembles from.

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.

Pass MATH 25 with a plan, not a cram

Upload your MATH 25 materials and Fennie generates a Daily Plan paced to your deadline — plus chat, flashcards, and quizzes built from the actual course content.

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