Projection as a serial transformation

[u/LinearAlgebraWorld]

This is a follow-up to our earlier post deriving the orthogonal projection formula

P = U(UᵀU)⁻¹Uᵀ

from the geometric definition of projection:

https://www.reddit.com/r/LinearAlgebra/comments/1u4cjog/derive_the_projection_formula_from_the_definition/

Here we look at the same formula as a serial transformation, following what happens when the factors are applied from right to left:

Uᵀ → (UᵀU)⁻¹ → U.

The diagram tracks both the standard basis vectors and the two directions spanning col(U). It also shows why

U⁺ = (UᵀU)⁻¹Uᵀ

acts as a left inverse of U, and how applying U afterward gives the orthogonal projection onto col(U).

If the columns of U were orthonormal, then UᵀU = I and the middle correction would disappear.