Make your hardest
courses learnable
Upload your lecture notes and textbooks. LearnForge indexes every page, then tutors you through them, asking the questions a good supervisor would, and citing the exact page every time.
- 7-Day Free Trial
- Works with PDFs, slides, images
- Your data stays yours
Students at 400+ institutions study here
Sessions that build intuition, not a highlight reel
Tell it what you already know. It starts just past that edge and works one question at a time, never dumping the answer, always naming the misconception when you miss.
- Adaptive difficulty from your demonstrated mastery, not a self-rating
- Every step traced to a page in your own material
- Ends with a synthesis prompt that forces you to connect the pieces
- Mastery per concept, tracked across sessions
- GlycolysisLearn
- The citric acid cycleLearn
- Electron transport chainLearn
- Chemiosmosis & ATP synthaseLearn
- Fermentation pathwaysLearn
Session built: 6 steps, starting at chemiosmosis, skipping glycolysis, which you've already demonstrated.
Switch between frontier models, all reading your books
Claude for dense proofs, Gemini for whole-course synthesis, GPT for structure. Same documents, same citations, one keystroke to swap.
- Clickable references that jump to the exact page
- Hybrid retrieval, keyword precision plus semantic recall
- Refuses to answer beyond your sources instead of inventing one
- Voice input, streaming responses, and Markdown export
Read the page and interrogate it at the same time
Document on the left, tutor on the right. Highlight anything and get it explained, simplified or exemplified without losing your place.
- Select-to-explain on any passage
- Answers scoped to what's actually on screen
- Reading progress saved per document
- Turn any paragraph into a flashcard in one click
6.2 Eigenspaces and multiplicity
For each eigenvalue λ, the eigenspace is the null space of A - λI. It contains every eigenvector for λ together with the zero vector.
The algebraic multiplicity of λ is how many times it appears as a root of the characteristic polynomial. The geometric multiplicity is the dimension of its eigenspace.
Geometric multiplicity is always at least one and never exceeds algebraic multiplicity. When they coincide for every eigenvalue, the matrix is diagonalizable.
A matrix where they differ is called defective, and no basis of eigenvectors exists.
Unlimited practice, generated from your syllabus
Pick topics, pick a difficulty, get a set. Distractors are built from genuine misconceptions in the source material, not random noise.
- Recall → understand → apply → analyse coverage
- Answer review with the exact passage that settles it
- Weak topics automatically fed back into your review queue
- Export to printable PDF for offline mock exams
Create a practice set
Questions generated from your own material, matched to your exam format.
Topics
Questions
12
Format
MCQ + short
Difficulty
Exam-level
The parts other tools skip
A chat box over a PDF is the easy 20%. Everything below is the other 80%. And it's what moves material from “I read it” to “I can reproduce it under exam conditions.”
Retrieval that keeps up with you
Hybrid BM25 + vector search with MMR de-duplication and a warm index cache. Typical retrieval over a full semester of notes: under 20 ms.
Pedagogy, not just prompts
Socratic questioning calibrated to what you've already demonstrated, then an FSRS-style scheduler that decides exactly when to bring each idea back.
Ask out loud
On-device speech recognition for questions and narration playback for summaries. Study while you walk, cook or commute.
Study rooms
Share a code, work the same deck together, see each other's answers live. Built on server-sent events so it works on any network.
Actually good on a phone
Every mode is designed mobile-first, thumb-reachable controls, swipeable cards, and reading view that doesn't collapse into a mess.
Your work, portable
Export decks to Anki or CSV, conversations to Markdown, quizzes to printable PDF. Nothing is locked in.
Students and tutors who stopped re-reading
I'd sworn off AI study tools, they all just summarise and make you feel productive. This one refuses to hand over the answer until you've reasoned toward it. That's the difference.
The citations are what sold me. Every claim links to the exact page in my own lecture notes, so I can check it in two seconds instead of trusting it blindly.
Four models in one place, all reading my textbooks. I cancelled two subscriptions the week I started using it.
The spacing scheduler is quietly the best part. I went from re-reading chapters to twelve minutes of review a day and my marks went up.
Why we built this
University is supposed to feel like progress, every term building toward something.
Mostly it feels like a blur: endless deadlines, late nights, and marks that never quite match the effort.
The problem isn't you.
It's that nobody ever taught you how to learn, how to test yourself before you feel ready, how to space the review, how to notice the gap between recognising a page and being able to reproduce it.
So we built the tool that does teach it, while you work through your own material.
Retrieval practice
Testing beats re-reading by a wide margin. Every mode is built around producing an answer, not recognising one.
Spaced repetition
An FSRS-style scheduler predicts when you're about to forget and surfaces the card right then.
Interleaving
Review queues mix topics deliberately, because blocked practice flatters you and mixed practice trains you.
Elaboration
Socratic prompts make you explain the mechanism, which is where the durable encoding happens.
Your notes are already there. Start using them.
One click to sign up. Upload a PDF and you'll have a tutored session, a deck and a practice set inside two minutes.