Vectors in Rⁿ

Geometry and algebra of linear maps, vectors, and matrices

A vector wears two hats at once. Numerically it is just an ordered list. [3, 1] means "3 then 1," and the order matters: [3, 1] is not [1, 3]. Geometrically that same list is an arrow: start at the origin, walk 3 right and 1 up, and the tip lands at the point the vector names.

Everything in linear algebra is built from this one object, so it pays to switch fluently between the two pictures: a vector is a list of coordinates and an arrow in space, and they are the same thing.

Think of a delivery drone leaving the depot. Its whole trip can be written as one arrow: [3, 4] means "fly 3 blocks east, then climb 4 storeys up," and the tip of that arrow is exactly where the package lands. The order of the slots is the route's instructions — east first, then up — so the list and the flight path are two names for the same journey.

Where this lives in MLVectors are the raw material of every model. A word embedding is a vector in R³⁰⁰⁰ (or larger); the weights feeding one neuron form a vector; the gradient that training follows is a vector pointing downhill in weight space. "Adding a gradient step to the weights" is the vector addition above: w ← w − η·g walks one arrow (the weights) along another (the scaled gradient).

▶ Vectors in Rⁿ

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