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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.

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.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Let V = C* be a complex inner product space (using the standard inner product for C*). Suppose that U1, U2 are subspaces of V and v1, V2 is an orthonormal basis for Uj and v3, V4 is an orthonormal basis for U2. Suppose that V = U1 O U2. Let P¡ be the orthogonal projection onto U1, and P2 be the orthogonal projection onto U2. Give the formula for P and P, (in terms of V1, V2, V3, V4). Then prove that a P + BP2 is a unitary matrix as long as |a| = |B| = 1 for a, ß E C.
Transcribed Image Text:Let V = C* be a complex inner product space (using the standard inner product for C*). Suppose that U1, U2 are subspaces of V and v1, V2 is an orthonormal basis for Uj and v3, V4 is an orthonormal basis for U2. Suppose that V = U1 O U2. Let P¡ be the orthogonal projection onto U1, and P2 be the orthogonal projection onto U2. Give the formula for P and P, (in terms of V1, V2, V3, V4). Then prove that a P + BP2 is a unitary matrix as long as |a| = |B| = 1 for a, ß E C.
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