Module 02 / §3.1–3.5
Vectors & spaces
Build intuition for vectors, distance, inner products, and projection.
11 lessons · Visual explanations · Guided experiments
02.1 / Explore → understand → practice
What can you build from two directions?
Explain how span, independence, and a basis fit together.
02.2 / Explore → understand → practice
Vectors as coordinates and displacements
Connect componentwise addition and scaling to a geometric displacement.
02.3 / Explore → understand → practice
Which sets keep every linear combination?
Test subspaces and explain when a sum has a unique decomposition.
02.4 / Explore → understand → practice
What must a linear map preserve?
Recognize a linear map and construct an isomorphism from bases.
02.5 / Explore → understand → practice
Why do matrix columns describe a map?
Recover a linear map from basis images and transpose products correctly.
02.6 / Explore → understand → practice
What disappears, and what can you reach?
Distinguish nullspace, range, row space, column space, and rank.
02.7 / Explore → understand → practice
How can distance define a limit?
Use the metric axioms and the epsilon–N definition of convergence.
02.8 / Explore → understand → practice
Does changing length change convergence?
Compare p-norms and explain the finite-dimensional norm equivalence statement.
02.9 / Explore → understand → practice
How does a product measure alignment?
Connect inner products, orthogonality, induced norms, and distance.
02.10 / Explore → understand → practice
How large can an inner product be?
Prove the orthogonal length identity and identify Cauchy–Schwarz equality cases.
02.11 / Explore → understand → practice
Why is the perpendicular foot the closest point?
Prove unique orthogonal decomposition and all eight projection properties.