Dot Product, Length, and Orthogonality

Dot product u * v =

u₁v₁ + u₂v₂ + … + u_nv_n

The length (or norm) of a vector v is

||v|| = sqrt(v₁² + v₂² + …. + v_n²)

A vector v with length 1 is called

a unit vector

A set of vectors {u₁, u₂, ….. , u_p} in Rⁿ is called an orthogonal set

if u_i * u_j = 0 when i ≠ j

An orthogonal basis of a subspace W of Rⁿ is a

basis of W that is an orthogonal set.

A set {u₁ , …, u_p} is called an orthonormal set if

it is an orthogonal set and u₁ , u₂ , …., u_p are unit vectors

Any orthogonal set of nonzero vectors can

be normalized to get an orthonormal set.

A basis that is orthonormal is called an

orthonormal basis

The orthogonal projection of y onto L is

y_L = ((y*u)/(u*u)) * u

Gram - Schmidt process to produce an orthogonal basis for W using a given basis for a subspace W.

v₁ = w₁

v₂ = w₂ - ((w₂ * v₁)/(v₁ * v₁))v₁ (formating is hard, the second part of the equation is literally just the orthogonal projection of y onto L with w₂ replacing y and v₁ replacing u)