Vector Operations

Linear Algebra (Matrices) · Axiom Academy

LESSON The Two Vector Operations Everything in a vector space is built from just two moves: adding two vectors tip-to-tail, and scaling one by a number. 1. Addition: Chain the Arrows Tip-to-Tail To add , slide so its tail sits at the tip of . The sum is the single arrow drawn straight from the origin to where you land. Algebraically, you just add matching components. Add the matching components, one row at a time — the diagonal of the parallelogram 2. Scalar Multiplication: Stretch, Shrink, or Flip To compute , multiply every component of by the number c . Geometrically the arrow stays on the same line through the origin — its length scales by |c| , and a negative c flips it to point the opposite way. The arrow grows: is twice as long, same direction. The arrow shortens: is half as long, same direction. The arrow reverses: has the same length but points the other way. The arrow vanishes to the zero vector at the origin. You've seen the two moves that generate every vector space — adding arrows tip-to-tail and scaling a single arrow. Scroll up to revisit either operation.

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