Lectures

Computing spent decades following exact recipes. Learning began when we replaced the recipe with a search, and the search runs on arrays of numbers.

What a dimension is, how vectors combine, and why the equation Ax = b runs through the whole course.

A matrix is a machine that moves every point of space at once. Its columns say everything about it, and some machines destroy information.

Every array carries a list of axes. Learning to read that list, and to say which axis should disappear, is most of what writing numerical code is.

The algorithm inside every solver. Turn a system into a staircase, read the answer from the bottom up, and discover that every step you took was a matrix.

Four readings of AB on one small example, why order matters and grouping does not, and what it takes to undo a matrix at all.

Split a matrix into simpler matrices and it starts talking: how many independent columns it really has, and exactly what elimination did to it.

Adding and scaling are the only two moves. Any place where they behave is a vector space, and the columns of a matrix build one of the most useful ones.

Add two solutions of Ax = 0 and you get another solution. Add two solutions of Ax = b and you do not. That gap is what makes one of them a subspace, and elimination hands you a basis for it.

One anchor plus the whole null space. Whether b fits at all, and how a single number, the rank, tells you how many answers to expect before you compute one.

One small matrix, four subspaces, four bases read straight off a single elimination, and the reason row rank always equals column rank.

Row space and null space meet at a right angle, and so does the other pair. Nothing perpendicular is missing, every reachable b comes from exactly one x, and that is where the pseudoinverse comes from.

When the target is out of reach, the best you can do is its shadow. The projection formula, the projection matrix, and why the miss is always perpendicular.

No line goes through all the data. Least squares picks the one that misses by the least, and it is the projection from the last chapter wearing different clothes.

Perpendicular unit columns make every formula in this course cheaper. How to build them from any basis, and the factorization that records the work.

One number says how much a matrix inflates or flattens space. Every rule, every formula, and every use of the determinant comes out of that.

Almost every direction gets turned when a matrix acts. A few survive and only stretch. Those directions run the long run, and their stretch factors shape every loss surface you will ever descend.