1. Let v ∈ C 4 be the vector given by v = (1, i, −1, −i). Find the matrix (with respect to the canonical

1. Let v ∈ C 4 be the vector given by v = (1, i, −1, −i). Find the matrix (with respect to the canonical basis on C 4 ) of the orthogonal projection P ∈ L(C 4 ) such that null(P) = {v} ⊥ . 2. Let U be the subspace of R 3 that coincides with the plane through the origin that is perpendicular to the vector n = (1, 1, 1) ∈ R3 . (a) Find an orthonormal basis for U. (b) Find the matrix (with respect to the canonical basis on R 3 ) of the orthogonal projection P ∈ L(R 3 ) onto U, i.e., such that range(P) = U.

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