Vectors and Linear Transformations Explained: 59 Linear Algebra Flashcards from Span to Jordan Form

Linear algebra is a vocabulary problem before it is a computation problem: most students can row-reduce a matrix, but exams trip them up with questions like "is this map injective?" or "what is the cokernel?" that hinges on a definition they half-remember. The deck Vectors & Linear Transformations (deck code BU8981, 59 cards) targets exactly that layer: every definition and theorem statement you need from vector spaces through eigenvalues, Jordan form and dual spaces.

Sample cards from the deck

Everything the 59 cards cover

• Vector space axioms and structure: vector addition, scalar multiplication, zero vector, additive inverse, scalars and fields, linear combination, linear independence, subspace, span, basis, dimension, coordinate vectors, hyperplane.
• Concrete examples: arrows in the plane and the parallelogram rule, R², coordinate space Fⁿ, complex numbers as a real vector space, function spaces, solution sets of homogeneous linear systems.
• Linear maps: definition, f(0) = 0, isomorphism, endomorphism, injective ⇔ ker(f) = {0}, surjective ⇔ im(f) = W, kernel, image, rank, nullity, rank-nullity theorem, linear extension from an independent set, cokernel W / im(f).
• Matrices: representation once bases are chosen, composition = matrix multiplication, determinant ≠ 0 ⇔ isomorphism, change of basis, rotation, reflection through the x-axis, uniform scaling kI, projection, shear.
• Eigentheory: eigenvector, eigenvalue, eigenspace, characteristic polynomial det(A − λI), Jordan canonical form, nilpotent (Tⁿ = 0) and idempotent (T² = T) operators.
• Constructions: quotient space V/W, direct sum, direct product, tensor product V ⊗ W, bilinear map, dual space V* = Hom(V, F), linear functional.
• Calculus connections: differentiation and integration as linear operators, and the kernel of a linear differential operator (f'' + f = 0 has kernel spanned by sin and cos).

Two tips that make the definitions stick

Read every "set" card as a subspace-closure test. Kernel, image, eigenspace, span and solution space are all defined as sets, and the deck's answers hide the same pattern: the set is closed under addition and scalar multiplication and contains 0. When you flip "Eigenspace", don't just recite "all eigenvectors for λ" — add "plus the zero vector, which is why it's a subspace". That extra clause is precisely what graders look for.

Draw the 2×2 cards before you flip them. Rotation, reflection, scaling, projection and shear are five cards in the deck. Sketch where the unit square goes under each matrix, then check the back. Once the picture is attached to the matrix, the abstract cards (injective, nilpotent, idempotent) get a concrete anchor: the projection (x, y) → (x, 0) is idempotent and not injective, and you can see it.

Pacing

Three sessions cover it: axioms through dimension (~20 cards), linear maps, rank-nullity and the five geometric matrices (~22), then eigentheory and the quotient/dual/tensor constructions (~17). After that, spaced repetition keeps a 59-card deck solid on a few minutes a day. Find it at flip-it.app with code BU8981.

If you're covering the broader math curriculum, the derivative rules in How to Learn Derivatives: Rules, Techniques & Applications pair well with the differentiation-as-linear-operator cards here.