40 free lessons · Worked examples and practice
Graduate math
Meet selected advanced ideas in analysis, algebra, geometry, probability, and fluid mathematics. These short introductions emphasize definitions, hypotheses, and the structure of an argument; they are starting points for deeper study.
Your study guide
What you will explore.
- Track definitions and theorem hypotheses
- Study advanced structures and analytical estimates
- Distinguish examples, general results, and research claims
Before you begin
A solid undergraduate foundation is expected, with prerequisites varying by topic. Review the relevant proof techniques and subject background before treating an advanced example as a general theorem.
Review UndergraduateHow to study these lessons
List every hypothesis and explain why it is needed. Reconstruct the argument, test a limiting case, and use the cited primary sources when a lesson discusses current research.
Work at your own pace. Open a hint when you need it, then try again before revealing the complete explanation.
Grade labels are starting-point guides; school sequences vary. University lessons introduce selected topics and assume the prerequisites listed on each page.
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650 guided practice chapters for Graduate.
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Your Graduate lesson collection.
The lessons appear in their suggested study order. Choose the first lesson for a starting point, or use the search and subject filters to review a particular skill. Read the prerequisites whenever you change topics.
Showing 12 of 40 matching lessons · All free
AlgebraGraduate · Extension 20 min
Recover a missing eigenvector with a Jordan chain
Use a generalized eigenvector when an eigenvalue lacks a full eigenbasis.
Read, practice, and reflectAlgebraGraduate · Extension 20 min
Find the shortest polynomial that annihilates a matrix
Distinguish a minimal polynomial from a characteristic polynomial.
Read, practice, and reflectAlgebraGraduate · Extension 20 min
Identify linear measurements as dual vectors
Construct a dual basis that extracts nonstandard coordinates.
Read, practice, and reflectAlgebraGraduate · Extension 20 min
Build a bilinear interaction from elementary tensors
Expand an elementary tensor in a product basis.
Read, practice, and reflectAlgebraGraduate · Extension 20 min
Check whether cosets support a group operation
Use conjugation to test normality before defining a quotient group.
Read, practice, and reflectAlgebraGraduate · Extension 20 min
Constrain subgroups using Sylow counting
Combine divisibility and congruence conditions on Sylow subgroup counts.
Read, practice, and reflectAlgebraGraduate · Extension 20 min
Use an ideal to make quotient multiplication consistent
Connect polynomial evaluation with a quotient ring.
Read, practice, and reflectAlgebraGraduate · Extension 20 min
Find the symmetries of a quadratic field extension
Identify automorphisms that fix the rational base field.
Read, practice, and reflectAlgebraGraduate · Extension 20 min
See a linear operator as a polynomial-module action
Encode repeated operator application through multiplication by a formal variable.
Read, practice, and reflectAlgebraGraduate · Extension 20 min
Average a symmetry action to extract invariant vectors
Construct a projection by averaging a finite group representation.
Read, practice, and reflectGeometryGraduate · Extension 20 min
Measure loop winding through a covering coordinate
Represent the fundamental group of a circle by integer winding.
Read, practice, and reflectGeometryGraduate · Extension 20 min
Count sheets of a circle covering
Relate a covering's degree to its action on fundamental groups.
Read, practice, and reflect