Let V = C* be a complex inner product space (using the standard inner product for C4). Suppose that U1,U2 are subspaces of V and v1, V2 is an orthonormal basis for U and v3, V4 is an orthonormal basis for U2. Suppose that V = U1 ÐU2. Let P¡ be the orthogonal projection onto U1, and P2 be the orthogonal projection onto U2. Give the formula for P and P2 (in terms of V1, V2, V3, V4). Then prove that a P¡ + ßP2 is a unitary matrix as long as |a| = |B| = 1 for a, ß E C.

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