### Let W R3 Be The Plane In R3 Which Has Equation X Y 2z 0 Let T Be The Orthogonal

1. Let W ⊂ R3 be the plane in R3 which has equation x − y + 2z = 0.

Let T be the orthogonal projection onto W , i.e. for any vector v ∈ R3, T (v) = projW (v) is the orthogonal projecton of v onto W.

(a) Find an orthogonal basis B1 for W and a basis for B2 for W⊥.

(b) Let B be the basis of R3 which is the union of B1 and B2. Find [T]B←B and verify that

[T]B←B ◦[T]B←B =[T]B←B

(c) Let E be the standard basis of R3. Find the change of basis matrices PE←B and PB←E

(d) Find [T]E←E using the change of basis formula.