“I can't trust ChatGPT math.”
Verification
It's confident either way. Ask twice, get two verdicts, and you still don't know if your answer was right.
First course: Linear Algebra for AI/ML
Master every branch.
Never lose the forest.
Rigorous math for STEM students working toward an exam, and for professionals rebuilding the foundations behind engineering, statistics, AI, and everything past the textbook. Graded like a great TA, not a hallucinating AI, always tied back to why it matters.
- Deterministic grading
- Every answer checked by computer algebra. No language model ever decides right or wrong.
- Expert-written
- Lessons, problems, and wrong-answer diagnoses reviewed by working mathematicians before they ship.
- Paired with code
- Every lesson ships with the Python that builds what you just learned, and a visualization wired to it.
First 3 lessons free, full grading included. No credit card to look around.
See how everything connects
No concept stands alone. Click any connection to see exactly what one idea hands to the next, all the way to where it lands in ML.
Click any dot on a line between two cards.
Content is cheap
You don't have an explanation problem. You have a verification problem.
Explanations are everywhere: 3Blue1Brown, a textbook, ChatGPT standing by for whatever's left. What none of them can do is tell you, reliably, whether the answer sitting in front of you is actually correct.
“I don't know what to study next.”
Structure
No path, and no honest answer for what's actually safe to skip.
“I don't know how this connects to the bigger picture.”
Motivation
A topic in isolation is trivia. You never see what it feeds, or what falls apart without it.
“I have no idea when I'll actually finish.”
Accountability
No progress model. No finish line in sight.
Why not just ask ChatGPT? Because a chatbot can explain the same concept five different ways, but it can't reliably tell you whether your specific answer is right. Ask it to grade the same work twice, phrased two different ways, and you can get two different verdicts. Language models are fluent, not deterministic. That's not a trust problem you can prompt your way out of. It's why the verdict here never comes from a language model at all.
Determinism first
The verdict is never AI. Only the sentence explaining it is.
Work it out your way: on paper, in a notebook, on a whiteboard, however it actually clicks for you. When you're ready, enter your final answer into a real equation editor, no LaTeX required, and our Deterministic Grader checks it instantly: real computer algebra, not a language model's best guess. Get it almost right, and we don't just say “wrong”: we tell you exactly which mistake you made, in plain English, using your own numbers. And soon, you'll be able to scan in your actual working and get it checked too, margin notes and all.
Real math decides. AI only explains.
Linear Algebra for AI/ML › Practice: Determinants
Question 3 of 8
What is det(A) for the matrix below?
A
12
B
14
Your answer
C
10
Correct answer
D
−10
Not quite: it looks like you added instead of subtracting. det(A) = ad − bc = (3 × 4) − (1 × 2) = 10.
Where AI fits, and where it doesn't
We use AI deliberately, and we're precise about where.
AI personalizes
It builds your path
An adaptive diagnostic reads what you already know, and AI shapes a learning path around it: what to study next, what you can safely skip, what to shore up first.
AI interprets
It explains the verdict
When you're wrong, AI turns the grader's finding into plain English, using your own numbers, the way a good TA talks you through it instead of handing back a red X.
AI never decides
It doesn't grade you
Right or wrong is settled by computer algebra: deterministic, equivalence-aware, the same answer every time. The math decides. The AI only explains the decision.
Personalized Learning Paths
Know what to learn next, and why it matters.
An adaptive diagnostic works out what you actually know. AI then shapes a learning path around it: a live map showing exactly where you stand, what to study next, and how every topic connects to the ones around it. See a prerequisite you could skip? Skip it. Feeling shaky on one? Do it first. You don't just know what to learn. You know why you're learning it, and how it fits the bigger picture.
Linear Algebra for AI/ML › Your Learning Path
You are here
Eigenvalues & Eigenvectors
Click any concept or connection in the tree to explore it.
you are here
prerequisite
Build to understand
Every lesson is paired with code.
“What I cannot build, I do not understand.”
— Richard Feynman
So you don't stop at the symbols. Every lesson ships with the code that builds the thing you just learned, and a live visualization wired to it. Move the math, watch the picture move, read the exact lines that produced it. Understanding you can run is understanding you actually have.
Linear Algebra for AI/ML › Exercise: Linear Combinations
Exercise 2 of 6
Reach the target vector
Every vector you can build from v1 and v2 is some a v1 + b v2: scale each one, add them tip to tail. That's the whole idea of a linear combination, and it's the machinery underneath span, basis, and everything after.
Given v1 = (2, 1) and v2 = (−1, 2), find the scalars a and b with a v1 + b v2 = (3, 4).
Drag the tip of w on the canvas, or move the scalars, and watch the code on the right change with it. Same numbers, same picture.
Code
Relevance, not just theory
Never resets
Know exactly why it matters
A topic only earns a real-world tag when an expert can point to where it's actually used: eigendecomposition in PCA and vibration analysis, dot products in physics and attention mechanisms, derivatives in every optimization problem you'll ever write. No hand-waving, no forced connections.
Progress that never resets
Test out of anything, any time, in one short proof-of-mastery set. Slip up? One extra review item. Never a reset: a bad Tuesday shouldn't cost you a month.
The Arena
Where LearnTree actually sits
Khanmigo tops out around AP depth. Math Academy is rigorous, but procedural: all drills, thin explanation. LearnTree is the university-rigor, self-paced option growing in the gap between them.
Khanmigo
K–12 depth, not university math
$4/mo
LearnTree
university rigor, deterministic grading, your pace
$20/mo
ALEKS
same diagnostic idea, resets your progress on a slip
~$20/mo
Brilliant
beautifully interactive, too shallow for rigor
$13–28/mo
Math Academy
closest rival: rigorous, but procedural and thin on explanation
$49/mo
And once a branch has grown, it's yours. We never delete your progress.
Experts, not scrapers
There's an expert behind every lesson.
Nothing ships until someone who actually knows the subject has signed off on it: the explanation, the problems, and the diagnosis you get when you're wrong. Curated for quality, one lesson at a time.
Expert-curated
Every lesson, problem family, and wrong-answer diagnosis is written and reviewed by working mathematicians and practitioners before it ships. AI drafts and generates. It never authors from scratch.
The problem bank
50,000+
Unique, computer-algebra-verified problems, generated from expert-authored seeds. No two students see quite the same version of a question, and no old answer key gets you very far.
One course at a time
Linear Algebra is just the first course.
A new course roughly every 6–10 weeks, each one unlocking more of what you can actually build.
Available at launch
Linear Algebra for AI/ML
Eigenvectors, transformations, PCA, and more.
Next
Calculus 1
Derivatives & gradients: the foundation of every optimization problem, from engineering to machine learning.
Calculus 2
Integrals & series: essential for physics, statistics, and probability.
Calculus 3
Multivariable calculus: the geometry behind fields, forces, and optimization.
Differential Equations
Continuous-time systems: from circuits and population models to neural networks.
Plus free precalculus and trig refreshers for anyone who needs to warm up first.
Pricing
What it costs
Monthly
$20/mo
Cancel anytime.
Semester pass (recommended)
$55/~4mo
Matches how a course actually gets used.
First 3 lessons are free, with full grading enabled, not a stripped demo.
Stop guessing if you got it right.
Reserve your seat for Linear Algebra for AI/ML, the first course on the tree. We'll email you the moment enrollment opens.
No credit card, no spam. Just one email, when we launch.