Question

If W is a subspace of Rⁿ with an orthonormal basis u₁, u₂, . . . , u_k, if x ∈ Rⁿ, and if

projW (x) = (x • u₁)u₁ + (x • u₂)u₂ + · · · + (x • u_k)u_k,

then x − projW ^(x) is orthogonal to every element of W.

(Please show that x − projW ^(x) is orthogonal to each u_j for 1 ≤ j ≤ k)

u,x are vectors.

Answer 1

Let W be a subspace of Rⁿ with an orthonormal basis u₁, u₂, . . . , u_k. Let x ∈ Rⁿ .Now, since { u₁, u₂, . . . , u_k } is an orthonormal basis for W, hence u_i.u_j = 0 if i ≠ j. and u_i.u_i= 1 for each 1 ≤ i ≤ k .

Further, since proj _W (x) = x- (x . u₁)u₁ + (x . u₂)u₂ + · · · + (x . u_k)u_k, hence x − proj _W (x) = x- (x . u₁)u₁ + (x . u₂)u₂ + · · · +(x . u_k)u_k so that (x − proj _W (x)). u_i = (x- (x • u₁)u₁+(x • u₂)u₂ + · · · + (x . u_k)u_k) . u_i = x.u_i – ((x . u₁)u₁ + (x . u₂)u₂ + · · · + (x . u_k)u_k).u_i = x.u_i – [ (x . u₁)u₁.u_i + (x . u₂)u₂.u_i +…+ (x . u_k)u_k.u_i ] = x.u_i – (x.u_i)(u_i.u_i) (as u_i. u_j = 0 if i ≠ j )= x.u_i – (x.u_i) ( as u_i.u_i = 1 for each 1 ≤ i ≤ k since all the u_i s are orthonormal vectors) .

Thus, (x − proj _W (x)).u_i = 0 for each i , 1 ≤ i ≤ k. Hence x − proj _W (x) is orthogonal to each u_j for 1 ≤ j ≤ k.