Be able to read a matrix as a thing that moves vectors, say what it stretches and what it destroys, and know what the common decompositions are actually telling you

Matrices as Maps

A matrix is not a grid of numbers to be memorised but a map that takes vectors somewhere. This course covers what the columns mean, why order matters, what rank throws away, and what eigenvectors and singular values are for.

8 lessons, written and corrected before you arrived. Reading them here needs no account. The first reads the whole way through; the others open and then stop, because a page nobody owns cannot tell who is reading it. Starting the course gives you your own copy, where every idea has problems standing under it and you can ask about any sentence.

Start reading

01Not a Grid, a VerbA matrix is a machine that takes a vector and returns a different one, and the only thing that makes it special is that it respects addition and scaling.
02Each Column Is a Destinationopening onlyThe columns of a matrix are the places the axes land, which turns reading a matrix into looking at a short list of destinations and writing one into a design decision.
03Why the Multiplication Rule Looks Like Thatopening onlyMatrix multiplication is doing one map and then another, which forces the strange entry rule, explains why order matters, and makes the cost of a chain a real decision.
04The Shadow and the Thingopening onlyEvery map reaches a limited set of outputs and destroys a definite set of inputs, and the two sizes add up to the input dimension, which is the whole bookkeeping of rank.
05Undo, If You Canopening onlyAn inverse is the map that undoes another one, it exists only when nothing was destroyed and everything is reachable, and solving a system is never the same job as building one.
06The Directions That Only Stretchopening onlySome directions come out of a map pointing exactly where they went in, only longer or shorter, and finding them turns repeated application of a map into simple arithmetic.
07Every Map Is a Rotation, a Stretch and a Rotationopening onlyEvery matrix without exception turns out to be a rotation followed by stretching along perpendicular axes followed by another rotation, and the stretches are the numbers that matter.
08When You Cannot Hit the Targetopening onlyMost real systems ask for something the map cannot produce, so the honest answer is the closest reachable point, and that one idea is the whole of fitting.