Lesson 1 of 10Article16 minFREE
Vectors: Geometry & Algebra
A vector packs magnitude and direction. In coordinates, v = ⟨v₁, …, vₙ⟩ lives in ℝⁿ. Addition is component-wise; scalar multiplication stretches or flips.
Arrows that add
A vector packs magnitude and direction. In coordinates, v = ⟨v₁, …, vₙ⟩ lives in ℝⁿ. Addition is component-wise; scalar multiplication stretches or flips.
- u·v = 0 ⇒ orthogonal (perpendicular in ℝ²/ℝ³).
- Projection of u onto v: proj_v u = ((u·v)/‖v‖²) v.
- Linear combinations: c₁v₁ + … + cₖvₖ — the heart of the subject.
Linear algebra is the mathematics of linear combinations. Everything else elaborates that idea.