UNI_FOUNDATION · Linear algebra · LA.ORTHOGONAL

Orthogonality and projection

Intuition

Understand first, compress later.

Linear algebra describes linear structure using vectors, matrices and transformations.

Switch representations

Orthogonal projectionp = 0.5·b

a = (2, 3)b = (5, 1)projection p = (2.5, 0.5)remainder a − p (⟂ b)

The shadow of a along b is p = 0.5·b = (2.5, 0.5). What is left over, a − p = (−0.5, 2.5), is perpendicular to b — that is what makes p the closest point of the line through b.

Formal view

You can orthogonality and projection when you can recognise the underlying objects, select valid rules and check each step for equivalence or logical implication.

Why does it work?

This skill sits on a prerequisite network. If a step is unstable, repair the missing prerequisite instead of merely doing more items of the same type.