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UVA MATH 3351: Elementary Linear Algebra

MATH 3351 is the math department's linear algebra course — matrices and row operations, vector spaces and bases, orthogonality, linear transformations, and eigenvalues — with deliberate emphasis on theory and abstract argument rather than pure computation. Credit isn't given for both MATH 3350 and 3351.

Fennie is independent and not affiliated with University of Virginia. This is an unofficial study guide.

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What makes it hard

The course pivots from computation to abstraction mid-stream: row reduction is mechanical, then vector spaces, subspaces, and linear independence demand definition-based reasoning and short proofs. Students expecting a matrix-arithmetic course hit the abstraction wall around the vector space unit, and exam questions test whether you can argue from definitions, not just calculate.

What you'll cover

  • • Matrices and elementary row operations
  • • Vector spaces, subspaces, and bases
  • • Linear independence and dimension
  • • Inner products and Gram-Schmidt
  • • Linear transformations and change of basis
  • • Eigenvalues, eigenvectors, and symmetric matrices

The MATH 3351 study guide

How to study for UVA MATH 3351, step by step.

  1. 1

    Learn every definition cold, immediately

    Span, independence, basis, subspace — exam arguments are built from definitions verbatim. Being able to state each one precisely, with an example and a non-example, is the course's real entry fee.

  2. 2

    Make row reduction free, then move on

    Gaussian elimination underlies half the computations and should cost no thought. Drill it early so your attention is available for the abstraction, which is what's actually graded.

  3. 3

    Practice the short proofs the course expects

    Show a set is a subspace; prove vectors are independent; verify a map is linear. These follow patterns — argue from the definition, line by line — and pattern fluency comes from doing dozens, not reading them.

  4. 4

    Connect every concept back to systems of equations

    Rank, null space, independence, and invertibility are all statements about solutions to Ax = b. Keeping that one thread visible turns the theorem lists into one coherent story — which is how exams test them.

  5. 5

    Translate between the three languages

    Matrices, linear transformations, and vector-space statements say the same things three ways. Practice converting any claim between them; exam questions are routinely just translation tests.

  6. 6

    Pace the abstraction with Fennie

    Upload your MATH 3351 syllabus and Fennie's Daily Plan spaces definition work and proof practice ahead of each exam, with flashcards for the definitions and quizzes generated from your actual course materials. Free to start.

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How Fennie helps with MATH 3351

Fennie's Daily Plans pace MATH 3351's pivot from computation to abstraction, spacing definition mastery and short-proof practice ahead of each exam. Chat tests whether you can argue from definitions — the skill 3351 actually grades — and flashcards keep span, basis, and dimension precise instead of approximately remembered.

FAQ

Is MATH 3351 at UVA hard?

Harder than students expect, because it's a theory course wearing a computation course's name. Row reduction is the easy part; vector spaces, independence arguments, and short proofs are what exams grade. Students who memorize definitions precisely and practice arguing from them do well.

What's the difference between MATH 3351 and MATH 3350?

MATH 3351 emphasizes theory and abstract argument; MATH 3350 (Applied Linear Algebra) leans computational and applied. Credit is given for only one. Math majors and the theoretically inclined take 3351 — check which one your program actually wants before enrolling.

What do I need before MATH 3351?

MATH 1320 is the formal prerequisite, but the real preparation is tolerance for definition-based reasoning — closer to discrete math than to calculus. If you've never written a short proof, expect the vector space unit to be where the adjustment happens.

Pass MATH 3351 with a plan, not a cram

Upload your MATH 3351 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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