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The Only Two Moves: Add and Scale

Almost everything in linear algebra is built from just two operations on [[vector|vectors]]: adding them tip-to-tail, and multiplying by a [[scalar|scalar]] to stretch, shrink, or flip. Master these two and the rest follows.

Move one: addition, tip to tail

To add two vectors, follow the first arrow, then start the second arrow where the first one ended. The single arrow from your start to your final stop is the sum. In list form it's even easier: just add the matching numbers.

(3, 1) + (1, 2) = (3+1, 1+2) = (4, 3)
Add coordinate by coordinate — order doesn't matter.

Move two: scaling

Multiplying a vector by a scalar keeps its direction but changes its length. Multiply by 2 and the arrow doubles. Multiply by 0.5 and it shrinks by half. Multiply by -1 and it flips to point the opposite way.

2 * (4, 1) = (8, 2)
0.5 * (4, 1) = (2, 0.5)
-1 * (4, 1) = (-4, -1)
Scaling multiplies every coordinate by the same number.

Why just these two?

Here's the quiet payoff: combine adding and scaling and you get a linear combination — scale a few vectors, then add the results. Every big idea ahead, from spans to matrices to machine learning, is built on top of this one move.