The experiment notebook
Change something.
See what follows.
Guided experiments connect a picture to a mathematical idea. Open a lesson to explore the controls, work an example, and test your prediction.
Guided experiment
Numbers that tell a story
Fractions, powers, roots, and logarithms are different ways to describe scale.
Guided experiment
Functions are machines
Follow an input through simple rules, then connect the rules together.
Guided experiment
Give a point an address
Build a coordinate from directions you can see, then turn the view in three dimensions.
Guided experiment
Read symbols as sentences
Turn sets, quantifiers, sums, and products into instructions you can follow.
Guided experiment
Change, one small step at a time
See derivatives as local slopes and integrals as accumulated contributions.
Guided experiment
How do you know it is true?
Separate a definition, an example, and a proof—and learn to break a false claim.
Guided experiment
The math behind a prediction
Meet the three toolkits—and learn what this course and its source actually cover.
Guided experiment
A field guide to the symbols
Translate every entry in the source’s notation table into words and a small example.
Guided experiment
Know what kind of thing you have
Scalars, vectors, matrices, and random variables play different roles—even when all of them involve numbers.
Guided experiment
Let dimensions catch mistakes
Before calculating, check whether the pieces can fit together.
Guided experiment
What can you build from two directions?
Explain how span, independence, and a basis fit together.
Guided experiment
Vectors as coordinates and displacements
Connect componentwise addition and scaling to a geometric displacement.
Guided experiment
Which sets keep every linear combination?
Test subspaces and explain when a sum has a unique decomposition.
Guided experiment
What must a linear map preserve?
Recognize a linear map and construct an isomorphism from bases.
Guided experiment
Why do matrix columns describe a map?
Recover a linear map from basis images and transpose products correctly.
Guided experiment
What disappears, and what can you reach?
Distinguish nullspace, range, row space, column space, and rank.
Guided experiment
How can distance define a limit?
Use the metric axioms and the epsilon–N definition of convergence.
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Does changing length change convergence?
Compare p-norms and explain the finite-dimensional norm equivalence statement.
Guided experiment
How does a product measure alignment?
Connect inner products, orthogonality, induced norms, and distance.
Guided experiment
How large can an inner product be?
Prove the orthogonal length identity and identify Cauchy–Schwarz equality cases.
Guided experiment
Why is the perpendicular foot the closest point?
Prove unique orthogonal decomposition and all eight projection properties.
Guided experiment
Which directions stay on the same line?
Identify eigenvectors and track eigenvalues under shifts, inverses, and powers.
Guided experiment
Which changes leave the diagonal sum alone?
Use trace linearity and cyclic invariance without assuming arbitrary reordering.
Guided experiment
When does a transformation flatten area?
Use the five determinant identities and distinguish area scale from length scale.
Guided experiment
Which matrices preserve length and angle?
Derive orthogonal matrix identities and prove preservation of inner products.
Guided experiment
Can a symmetric map be split into three simple steps?
Derive the spectral decomposition from an orthonormal eigenbasis.
Guided experiment
Which direction gives the largest quadratic value?
Prove the Rayleigh bounds and their equality conditions, including repeated eigenvalues.
Guided experiment
When is every nonzero direction uphill?
Prove the eigenvalue criteria and explain why a positive diagonal shift removes singularity.
Guided experiment
Why does a larger eigenvalue make a shorter ellipse axis?
Derive quadratic level sets from a sphere and identify their eigenvector directions.
Guided experiment
What does a matrix stretch?
Follow orthogonal changes of coordinates and singular-value scaling, including rectangular matrices.
Guided experiment
Which part of an input survives?
Decompose inputs and outputs into the four fundamental subspaces.
Guided experiment
How much can a matrix amplify a vector?
Compare induced norms, Frobenius size, and orthogonal invariance.
Guided experiment
How much can you discard?
Use truncated SVD and prove its spectral-norm optimality.
Guided experiment
Can you undo a map that loses information?
Construct the Moore–Penrose pseudoinverse and verify its four conditions.
Guided experiment
How do columns and outer products build a matrix?
Translate matrix products and quadratic forms into sums of entries.
Guided experiment
Where are you allowed to look for a minimum?
Distinguish feasible, local, global, and strict minima.
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Which direction changes the value fastest?
Read the gradient as a column of partial derivatives and a local direction of steepest change.
Guided experiment
What linear map describes a nonlinear change nearby?
Match Jacobian rows to outputs and columns to inputs.
Guided experiment
How does curvature change a step?
Use the Hessian to measure directional curvature and compare gradient and Newton steps.
Guided experiment
Why does a quadratic gradient contain a transpose?
Derive linear and quadratic gradients, including nonsymmetric matrices.
Guided experiment
How does a change travel through a composition?
Multiply Jacobians in the correct order and propagate a scalar gradient.
Guided experiment
How accurate is the local picture?
Distinguish first-order, integral-gradient, and second-order Taylor formulas.
Guided experiment
When does a stationary point really minimize?
Prove necessary and sufficient local-minimum conditions and test their limits.
Module 2 · coordinate scaling
Stretch a vector
A positive scale preserves the direction.