Let W be a subspace of Rn Define the orthogonal complement o

Let W be a subspace of R^n. Define the orthogonal complement of W, denoted W Slow that W is a sub space of R^n.

Solution

a.     If W is a subspace of Rⁿ , the orthogonal complement of W in Rⁿ   is the set W of vectors u such that u is orthogonal to all vectors in W.

b. Let u₁, u₂ be two arbitrary vectors in W and let w be an arbitrary vector in W. Then w. u₁ = 0 and w. u₂ = 0 . Also w. (u₁ + u₂ ) = w. u₁ + w. u₂ = 0+ 0 = 0. Thus, W is closed under vector addition. Now, let be an arbitrary scalar in R. Then w. u₁ = (w. u₁) = *0 = 0. Thus W is closed under scalar multiplication. Also, when = 0, the zero vector belongs to W. Thus, W is a subspace of Rⁿ