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Polynomial coordinates and infinite-dimensional flavor
Coefficients of a polynomial act like coordinates with respect to the basis 1, x, x^2, ldots. Each...
Abstraction trades pictures for generality
Textbooks phrase linear algebra axiom-first so theorems apply to every certified vector space [@2]....
Same words, new habitats: eigenfunctions, inner products
Eigenvectors become eigenfunctions when the vector space is a function space. Dots become inner...
Series close: concrete roots, abstract canopy
The Essence playlist began with arrows in the plane and ends with permission to reuse every tool on...
Vectors as displacements, not "a point you happen to draw"
The chapter opens with a deliberately informal picture: a vector is an arrow encoding how the world...
Numeric coordinates: bookkeeping layered on top of geometry
Once an origin and perpendicular axes are fixed, every displacement acquires a pair of numbers...
Coordinates as linear combinations of basis arrows
Writing (x,y) as xhatmathbfi+yhatmathbfj reveals the secret grammar: coordinates list the scalars...
Chapter close: language hygiene before transformations
The narrative tightens vocabulary: distinguish vectors , points , coordinates , and bases before...
Grids become parallelepiped lattices, still linear
Chapter 4 made two-dimensional grids bend into parallelogram lattices while preserving the linear...
Columns still list where $\hat{\mathbf{i}},\hat{\mathbf{j}},\hat{\mathbf{k}}$ land
The column recipe from two dimensions survives unchanged: column one is Ahatmathbfi, column two is...
Orientation: right-hand rule survives or flips
Orientation is about whether a small right-handed tripod of vectors keeps that handedness after the...
Rank in three dimensions: lines, planes, volumes
Rank counts how many independent directions survive in the output. Rank 1 collapses all of...