Matrix
A matrix is a picture of what happens to the grid
Four numbers in a box is not an object anyone has intuitions about. The same four numbers, shown as an instruction for redrawing the plane, are.
Seven bridges, and the invention of throwing things away
Euler solved a puzzle about a Prussian city by deleting the city. What survived the deletion was a new branch of mathematics.
The dot product is a shadow
Multiply the matching coordinates and add them up. That rule explains nothing, and it hides the fact that the answer is a length — how far one arrow reaches along another, times how long that other one is.
The directions a map leaves alone
Almost every arrow comes out of a transformation pointing somewhere else. A few come out pointing exactly where they went in, only longer or shorter. Those few decide nearly everything the map does.
Named alongside it
The objects these essays reach for when they reach for this one.
BasisDeterminantOrthogonalityAbstractionAreaCharacteristic polynomialCosineDegreeDiagonalisationDot productEigenvalueEigenvector