Alright, I’ll simplify and shorten the theorems to match the style of your cheatsheet.
Orthogonality
Definition
n ∈ ℝ n is orthogonal to subspace V of ℝ n if n • v = 0 for all v in V. (n ⊥ V)
Theorem
If S = {u 1 , u 2 , …, u k } spans V, then w is orthogonal to V iff w • u i = 0 for all i.
Theorem
If S = {u 1 , u 2 , …, u k } spans V and A = (u 1 u 2 … u k ), then
w ⊥ V ⇔ w ∈ Null(A T ).
Orthogonal Complement
Definition
V ⊥ = { w ∈ ℝ n | w • v = 0 for all v in V }.
Theorem
If S = {u 1 , u 2 , …, u k } spans V and A = (u 1 u 2 … u k ), then
V ⊥ = Null(A T ).