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.
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 rather than only 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
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
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
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
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
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.
Today
Today's MATH 3351 plan
What a Fennie Daily Plan looks like for MATH 3351. Yours is built from your own syllabus and adapts every day to your deadlines and progress.
First plan free, no card required. Fennie is independent and unaffiliated with your school.
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.
More UVA courses
MATH 1310: Calculus I
MATH 1310 is the College of Arts & Sciences' Calculus I: limits, derivatives, applications of differentiation, and the beginnings of integration. It serves math, science, economics, and pre-health tracks. Engineering students take the parallel APMA sequence instead.
MATH 1320: Calculus II
MATH 1320 continues the College's calculus sequence with techniques of integration, applications of the integral, sequences and series, and parametric and polar topics. It's widely considered the harder half of the sequence and a prerequisite gateway for math, economics, and science tracks.
MATH 2310: Calculus III
MATH 2310 is multivariable calculus for the College: vectors, partial derivatives, multiple integrals, and the vector calculus capstones (line and surface integrals, Green's and Stokes' theorems). It's required or expected for math, physics, economics-quantitative, and CS-adjacent tracks.