Let W be a subspace of R", and let W- be the set of all vectors orthogonal to W. Show that W- is a subspace of R" 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 zi + z2 is orthogonal to u. What can you conclude about z¡ + z2? Why? c. Finish the proof that W- is a subspace of R".

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