Chapter 6.1, Problem 30E

Let W be a subspace of ℝⁿ, and let W^⊥ be the set of all vectors orthogonal to W. Show that W^⊥ is a subspace of ℝⁿ using the following steps.

a. Take z in W^⊥, and let u represent any element of W. Then z·u = 0. Take any scalar c and show that cz is orthogonal to u. (Since u was an arbitrary element of W, this will show that cz is in W^⊥.)

b. Take z₁ and z₂ in W^⊥, and let u be any element of W. Show that z₁ + z₂ is orthogonal to u. What can you conclude about z₁ + z₂? Why?

c. Finish the proof that W^⊥ is a subspace of ℝⁿ.