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Vector addition: head-to-tail choreography
To add mathbfu and mathbfv, draw mathbfu, place the tail of mathbfv at the tip of mathbfu, and the...
Geometry: eigen directions reveal intrinsic stretching
Reading columns shows where basis vectors land, but eigenvectors summarize natural axes of strain...
Many things are "vectorish"
Arrows in the plane, tuples of numbers, polynomials, and signals can all reuse linear algebra...
Coordinates as translators between vectors and tuples
Once a basis is fixed, every vector gains a numeric fingerprint: the ordered coefficients. Changing...
Grids become parallelepiped lattices, still linear
Chapter 4 made two-dimensional grids bend into parallelogram lattices while preserving the linear...
Why depth needs nonlinear breaks
Before the algebra, the narrative asks for a hope about hidden layers: maybe a penultimate layer...
Alignment after pretraining
Pre-training imitates internet-scale text, including toxicity, bias, and unsafe instructions....
Bridge to transformer mechanisms
Language models before transformers often used RNNs : a hidden state updated serially along time,...
Non-commutativity: order usually matters
Even when both AB and BA exist, they often differ. Different pipelines deform space differently...
Composition: the matrix nearest the vector acts first
If S applies first then T, the composite on column vectors is TS, so (TS)mathbfx = T(Smathbfx)...
Guest bridge: from classical optimization to generative stacks
Welch Labs extends the playlist's visual vocabulary toward image and video synthesis. The same...
Training stability at scale
Training billion-parameter transformers is as much engineering as theory. AdamW decouples weight...