A square matrix A is idempotent if A? = A. Let V be the vector space of all 2 × 2 matrices with real entries. Let H be the set of all 2 x 2 idempotent matrices with real entries. Is H a subspace of the vector space V? 1. Does H contain the zero vector of V? H contains the zero vector of V 2. Is H closed under addition? If it is, enter CLOSED. If it is not, enter two matrices in H whose sum is not in H, using a comma [1 2] [5 separated list and syntax such as [[1,2], [3,4]], [[5,6], [7,8]] for the answer 3 4 (Hint: to show that H is not closed under addition, it is sufficient to find two idempotent matrices A and B such that (A+ B)² + (A + B).) [1.1].[0,0].[1.0].[1,1] 3. Is H closed under scalar multiplication? If it is, enter CLOSED. If it is not, enter a scalar in R and a matrix in H whose product is not in H, using a comma separated list and syntax such as 2, [[3,4], [5,6]] for the answer 2, . (Hint: to show that H is not closed under scalar multiplication, it is sufficient to find a real number r and an idempotent matrix A such that (rA)² # (rA).) (1,1),(0,0))

A square matrix A is idempotent if A? = A. Let V be the vector space of all 2 × 2 matrices with real entries. Let H be the set of all 2 x 2 idempotent matrices with real entries. Is H a subspace of the vector space V? 1. Does H contain the zero vector of V? H contains the zero vector of V 2. Is H closed under addition? If it is, enter CLOSED. If it is not, enter two matrices in H whose sum is not in H, using a comma [1 2] [5 separated list and syntax such as [[1,2], [3,4]], [[5,6], [7,8]] for the answer 3 4 (Hint: to show that H is not closed under addition, it is sufficient to find two idempotent matrices A and B such that (A+ B)² + (A + B).) [1.1].[0,0].[1.0].[1,1] 3. Is H closed under scalar multiplication? If it is, enter CLOSED. If it is not, enter a scalar in R and a matrix in H whose product is not in H, using a comma separated list and syntax such as 2, [[3,4], [5,6]] for the answer 2, . (Hint: to show that H is not closed under scalar multiplication, it is sufficient to find a real number r and an idempotent matrix A such that (rA)² # (rA).) (1,1),(0,0))

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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