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1. [10] Let ~x ∈ R n with ~x 6= ~0. For each ~y ∈ R...

1. [10] Let ~x ∈ R n with ~x 6= ~0. For each ~y ∈ R n , recall that perp~x(~y) = ~y − proj~x(~y).

(a) Show that perp~x(~y + ~z) = perp~x(~y) + perp~x(~z) for all ~y, ~z ∈ R n .

(b) Show that perp~x(t~y) = tperp~x(~y) for all ~y ∈ R n and t ∈ R.

(c) Show that perp~x(perp~x(~y)) = perp~x(~y) for all ~y ∈ R n

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