Harvard 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.

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What makes it hard

The course shifts from computation to abstraction midway: row reduction is mechanical, but linear independence, subspaces, and eigenvector reasoning require a conceptual leap many students don't see coming. The differential equations finale then stacks new material on top of whatever linear algebra didn't fully solidify.

What you'll cover

• Systems of linear equations and matrices
• Linear transformations
• Subspaces, basis, and dimension
• Eigenvalues and eigenvectors
• Orthogonality and least squares
• Differential equations and Fourier series

The MATH 21B study guide

How to study for Harvard MATH 21B, step by step.

1
Attach a concept to every computation
Don't just find eigenvalues — articulate what they tell you about the transformation. Math 21B exam questions live at the junction between computing and understanding, so practice both together from week one.
2
Slow down for the subspace weeks
Linear independence, span, basis, and dimension are where 21B's abstraction leap happens, and it doesn't announce itself. Give those weeks extra time and build small concrete examples for every definition.
3
Test yourself with true/false concept questions
Can a set of five vectors in R4 be independent? Does every matrix have an eigenvector? Conceptual quick-fire questions expose the gaps that pure computation practice hides.
4
Keep linear algebra warm into the differential equations finale
The ODE and Fourier material builds directly on eigenvalues and orthogonality. A weekly review of the earlier units prevents the end-of-semester stack-up the course is known for.
5
Front-load the hard weeks with Fennie
Upload the Math 21B syllabus and Fennie's Daily Plan concentrates conceptual review in the subspace and eigenvalue weeks where the course actually gets hard, with concept-to-computation quizzes generated from your real materials. Free to start.
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How Fennie helps with MATH 21B

Fennie's Daily Plans front-load conceptual review in the subspace and eigenvalue weeks where Math 21B actually gets hard. Chat through what an eigenvector means geometrically when the definitions feel circular, and quiz yourself on the concept-to-computation links the exams test.

FAQ

Is Math 21B harder than 21A?

Many students find 21B's abstraction harder than 21A's visualization, especially the subspace and eigenvalue material. The computational parts are straightforward; the concepts are the course.

Do I need Math 21B for CS at Harvard?

Linear algebra at the 21B level (or beyond) appears in the requirements or strong recommendations for CS and applied math tracks. Check current concentration requirements.

How do I study for Math 21B?

For every computation, attach the concept: don't just find eigenvalues, articulate what they tell you about the transformation. Exam questions live at exactly that junction.

Pass MATH 21B with a plan, not a cram

Upload your MATH 21B 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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