Section 21

Orthogonality and projections

intermediateviz

u ⊥ v ⟺ u \cdot v = 0

Characterizes perpendicular vectors.

intermediateviz

proj_{b} (a) = \frac{a \cdot b}{∥ b ∥ ^{2}} b

Projects a vector onto the direction of another.

intermediateviz

Q^{T} Q = I \Rightarrow Q^{- 1} = Q^{T}

Preserves lengths and angles.

advancedviz

u_{k} = v_{k} - j < k \sum proj_{u_{j}} (v_{k})

Builds an orthogonal or orthonormal basis from independent vectors.