The single idea that unlocks linear algebra: a matrix isn't a grid of numbers — it's a function that moves space. Every 2×2 matrix takes the plane and stretches, rotates, shears, or flips it, and you can read exactly what it does straight off its columns.
A matrix's columns are where î = (1, 0) and ĵ = (0, 1) land, and that fixes every other vector,
since each is a mix of the two. For [[a, b], [c, d]]: î → (a, c), ĵ →
(b, d). Drag the four numbers and watch the grid move.
The blue parallelogram is where the original unit square ends up. Its area is the
determinant, det = ad − bc. So the determinant has a concrete meaning: it's the factor
by which the transformation scales every area. A determinant of 3 triples areas; 0.5 halves them; a
negative determinant means space got flipped over (orientation reversed), like a mirror. And the
critical case — det = 0 — means the transformation squashes the whole plane onto a line (or a point):
it collapses a dimension. That's exactly when a matrix is singular and has no inverse, because once
you've flattened space you can't unambiguously un-flatten it. (This is why "det ≠ 0" is the test for an
invertible matrix and a unique solution to Ax = b.)
Matrix multiplication is just doing one transformation then another; an eigenvector is a special direction the transformation only stretches without rotating; PCA finds the directions of greatest stretch in your data's covariance. All of it rests on this picture of a matrix as a moving grid. When a neural net multiplies by a weight matrix, it's transforming its input space exactly like this — just in many more dimensions.
ad − bc is the area-scaling factor — negative means a
flip, and zero means space is collapsed onto a line (singular, non-invertible). Multiplication =
composing transformations; eigenvectors are the directions left un-rotated.Curated companions: 3Blue1Brown — Essence of Linear Algebra and Setosa — Eigenvectors.