(b) Recall that a real inner product space is a real vector space V together with which satisfies the following axioms. For all u, v, w e V and k E R: dtrce wat tlero.com a map (i) {u, v) = (v, u) (ii) (u, v + kw) = (u, v) + k(u, w) (iii) (u, u) > 0 and (u, u) = 0 iff u = 0 Also recall that given an inner product (·, ·), we defir the associated norm by ||u|| = V(u, u). Prove the generalized Pythagorean Theorem: If V is a real inner product space and u, v E are orthogonal, then ||u + v||? = ||u||² + ||v||?. V At each step in your proof, be sure to specify which of аplying. three inner product axioms (if any) you are (c) Let A and B be square matrices with entries in Z2. Prove that if A and B commute then (A + B)² = A² + B².

(b) Recall that a real inner product space is a real vector space V together with which satisfies the following axioms. For all u, v, w e V and k E R: dtrce wat tlero.com a map (i) {u, v) = (v, u) (ii) (u, v + kw) = (u, v) + k(u, w) (iii) (u, u) > 0 and (u, u) = 0 iff u = 0 Also recall that given an inner product (·, ·), we defir the associated norm by ||u|| = V(u, u). Prove the generalized Pythagorean Theorem: If V is a real inner product space and u, v E are orthogonal, then ||u + v||? = ||u||² + ||v||?. V At each step in your proof, be sure to specify which of аplying. three inner product axioms (if any) you are (c) Let A and B be square matrices with entries in Z2. Prove that if A and B commute then (A + B)² = A² + B².

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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