Question 5. Let V be a finite-dimensional inner product space, and let p, q: V→ V be two orthogonal projections. (a) Prove that the composition pq is an orthogonal projection if and only if pq = qp. (b) Prove that if pq is an orthogonal projection then ker pq = ker p + ker q.
Question 5. Let V be a finite-dimensional inner product space, and let p, q: V→ V be two orthogonal projections. (a) Prove that the composition pq is an orthogonal projection if and only if pq = qp. (b) Prove that if pq is an orthogonal projection then ker pq = ker p + ker q.
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter4: Vector Spaces
Section4.3: Subspaces Of Vector Spaces
Problem 49E
Question
![Question 5. Let V be a finite-dimensional inner product space, and let p, q: V→ V be two
orthogonal projections.
(a) Prove that the composition pq is an orthogonal projection if and only if pq = qp.
(b) Prove that if pq is an orthogonal projection then ker pq = ker p + ker q.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F645d1013-60f0-4983-bc35-5ec09c222357%2F7ed186cc-5722-43bf-9da1-b21eb3b45bb7%2Fuwfbe5k_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Question 5. Let V be a finite-dimensional inner product space, and let p, q: V→ V be two
orthogonal projections.
(a) Prove that the composition pq is an orthogonal projection if and only if pq = qp.
(b) Prove that if pq is an orthogonal projection then ker pq = ker p + ker q.
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