Please solve and answer in the same format as the question as it makes it easier to follow. Box answers and do not type it out and I will give you a thumbs up rating if answered correctly.Chapter 4.1 Question2 A square matrix A is nilpotent if A" = 0 for some positive integer n.Let V be the vector space of all 2 x 2 matrices with real entries. Let H be the set of all 2 x 2 nilpotentmatrices with real entries. Is H a subspace of the vector space V?1. Is H nonempty?H is nonempty ♥2. Is H closed under addition? If it is, enter CLOSED. If it is not, enter two matrices in H whose sum isnot in H, using a comma separated list and syntax such as [[1,2], [3,4]], [[5,6],[7,8]] for the: 1: 1. (Hint: to show that H is not closed under addition, it is sufficient to find two[7 8answer3 4nilpotent matrices A and B such that (A + B)" + 0 for all positive integers n.)3. Is H closed under scalar multiplication? If it is, enter CLOSED. If it is not, enter a scalar in R and amatrix in H whose product is not in H, using a comma separated list and syntax such as[3 472, [3,4], [5,6]] for the answer 2, : . (Hint: to show that H is not closed under scalar5 6multiplication, it is sufficient to find a real number r and a nilpotent matrix A such that (r A)" + 0 forall positive integers n.)4. Is H a subspace of the vector space V? You should be able to justify your answer by writing acomplete, coherent, and detailed proof based on your answers to parts 1-3.H is not a subspace of V v

Introduction: When a non-empty subset of a vector space is closed under addition and scalar…

A square matrix A is nilpotent if A" = 0 for some positive integer n. Let V be the vector space of all 2 × 2 matrices with real entries. Let H be the set of all 2 × 2 nilpotent matrices with real entries. Is H a subspace of the vector space V? 1. Is H nonempty?

A square matrix A is nilpotent if A" = 0 for some positive integer n. Let V be the vector space of all 2 × 2 matrices with real entries. Let H be the set of all 2 × 2 nilpotent matrices with real entries. Is H a subspace of the vector space V? 1. Is H nonempty?

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter5: Orthogonality

Section5.4: Orthogonal Diagonalization Of Symmetric Matrices

Problem 26EQ

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Please solve and answer in the same format as the question as it makes it easier to follow. Box answers and do not type it out and I will give you a thumbs up rating if answered correctly.

Chapter 4.1 Question2

A square matrix A is nilpotent if A" = 0 for some positive integer n.
Let V be the vector space of all 2 x 2 matrices with real entries. Let H be the set of all 2 x 2 nilpotent
matrices with real entries. Is H a subspace of the vector space V?
1. Is H nonempty?
H is nonempty ♥
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 separated list and syntax such as [[1,2], [3,4]], [[5,6],[7,8]] for the
: 1: 1. (Hint: to show that H is not closed under addition, it is sufficient to find two
[7 8
answer
3 4
nilpotent matrices A and B such that (A + B)" + 0 for all positive integers n.)
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
[3 47
2, [3,4], [5,6]] for the answer 2, : . (Hint: to show that H is not closed under scalar
5 6
multiplication, it is sufficient to find a real number r and a nilpotent matrix A such that (r A)" + 0 for
all positive integers n.)
4. Is H a subspace of the vector space V? You should be able to justify your answer by writing a
complete, coherent, and detailed proof based on your answers to parts 1-3.
H is not a subspace of V v

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