Essence of linear algebra (3Blue1Brown)
Intermediate
Mathematics
by Best
Vectors, linear maps, determinants, eigenvalues, and change of basis, taught through geometry. Sixteen modules with exercises and short animations, aligned with 3Blue1Brown's Essence of linear algebra.
University approvals: 1
(ZHAW - Zürcher Hochschule für Angewandte Wissenschaften: 1)
Coordinates, geometric arrows, span preview, and careful language about what "vector" names.
- Vectors as displacements, not "a point you happen to draw" Card
- Numeric coordinates: bookkeeping layered on top of geometry Card
- Vector addition: head-to-tail choreography Card
- Scalar multiplication: stretch, squeeze, and flip Card
- Coordinates as linear combinations of basis arrows Card
- Chapter close: language hygiene before transformations Card
Span, linear independence, bases, and coordinates as unique coefficient lists once a basis is fixed.
- Linear combinations: scale, then add, same recipe in every dimension Card
- Span: the flat region of everything reachable from a generating set Card
- Linear independence: no hidden redundancy in the generator list Card
- Basis: minimal spanning set, no junk directions, no missing directions Card
- Coordinates as translators between vectors and tuples Card
- Chapter synthesis: independence controls redundancy, span controls reach Card
Grids stay grids; columns record basis images; matrix-vector product combines those columns.
- Linearity: additivity and scaling, no translation of space Card
- Matrices in the standard basis: columns are destination arrows Card
- Matrix-vector multiplication: combine columns with input coefficients Card
- Examples: rotation, shear, projection, same rules, different columns Card
- Non-invertible maps collapse at least one direction Card
- Packing the chapter for matrix multiplication next time Card
Order of multiplication matches order of applying maps; column and row pictures of products.
- Composition: the matrix nearest the vector acts first Card
- Non-commutativity: order usually matters Card
- Column picture of $AB$: $A$ times each column of $B$ Card
- Row picture: rows of $A$ dot columns of $B$ Card
- Powers and repeated transformations Card
- Looking ahead: encode all sequential computation pipelines Card
$\mathbb{R}^3$ lattices: volume-thinking preview; reflections and shears.
- Grids become parallelepiped lattices, still linear Card
- Columns still list where $\hat{\mathbf{i}},\hat{\mathbf{j}},\hat{\mathbf{k}}$ land Card
- Orientation: right-hand rule survives or flips Card
- Rank in three dimensions: lines, planes, volumes Card
- Composing rotations and shears in animation pipelines Card
- Chapter stop: you are ready for determinant as volume scale Card
Signed area and volume scaling; zero map collapses dimension; multiplicativity.
- Determinant as volume scale factor for the column brick Card
- Orientation: sign of determinant Card
- Row operations and Gaussian intuition Card
- Invertibility test: $\det A\neq0$ for square matrices Card
- Geometric story before cofactor expansions Card
- Chapter synthesis: keep both absolute value and sign in mind Card
Image vs kernel; rank-nullity; solving $A\mathbf{x}=\mathbf{b}$ existence and uniqueness.
- Column space collects feasible outputs $\mathbf{b}$ Card
- Null space: differences of solutions; invisible inputs Card
- Invertibility: two-sided when square and full rank Card
- Computing inverses conceptually with row operations Card
- Rank-one updates intuition Card
- Full conceptual checklist before nonsquare focus Card
Tall vs wide maps; embeddings; projections between ambient spaces.
Projections, orthogonality, covectors, and the row picture of $A\mathbf{x}$.
Normal vector, area, right-hand rule; determinant mnemonic.
How $\det(T)$ mediates $T(\mathbf{a}\times\mathbf{b})$ vs $T(\mathbf{a})\times T(\mathbf{b})$.
- The naive cross-interchange identity usually fails Card
- Adjugate / cofactor matrix viewpoint (conceptual) Card
- Orthogonal maps split into rotations and reflections Card
- Computing examples without memorizing a long formula Card
- Edge cases with rank-deficient $T$ Card
- Bridge toward Cramer and signed volumes Card
Signed volumes recover coordinates of unknown inputs from output data and matrix columns.
Translate vectors; express the same linear map in Jennifer's language vs yours.
- Coordinates depend on basis; geometry does not Card
- Transformations in alternate coordinates sandwich Card
- Parsing the "backwards" feeling Card
- Trace and determinant stay similar Card
- Worked mental checklist for change of basis Card
- Lead-in: eigenbasis is the ultimate friendly coordinate change Card
Special directions only scaled; characteristic polynomial; diagonalization when possible.
- Eigenvector: stays on its span, only stretches or flips Card
- Characteristic polynomial: $\det(A-\lambda I)=0$ Card
- Eigenbasis and diagonalization Card
- Trace and determinant from eigenvalues (with multiplicity) Card
- Geometry: eigen directions reveal intrinsic stretching Card
- Bridge to the quick $2\times2$ eigenvalue trick Card
Read trace and determinant, then recover $2\times 2$ eigenvalues with $m \pm \sqrt{m^2-p}$.
Vectors as anything with consistent addition and scaling; axioms; functions as vectors.
- Many things are "vectorish" Card
- Linearity revisited: additivity + homogeneity Card
- Polynomial coordinates and infinite-dimensional flavor Card
- Abstraction trades pictures for generality Card
- Same words, new habitats: eigenfunctions, inner products Card
- Series close: concrete roots, abstract canopy Card