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 x 2 nilpotent matrices with real entries. Is H a subspace of the vector space V? 1. Does H contain the zero vector of V? choose 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 [12] [56] such as [[1,2], [3,4]], [[5,6], [7,8]] for the answer [³ ¾), § 9]. (Hint: 3 78 to show that H is not closed under addition, it is sufficient to find two nilpotent matrices A and B such that (A+B)" 0 for all positive integers n.) [34] 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, 5 6 (Hint: to show that H is not closed under scalar multiplication, it is sufficient to find a real number r and a nilpotent matrix A such that (rA)" #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. choose
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 x 2 nilpotent matrices with real entries. Is H a subspace of the vector space V? 1. Does H contain the zero vector of V? choose 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 [12] [56] such as [[1,2], [3,4]], [[5,6], [7,8]] for the answer [³ ¾), § 9]. (Hint: 3 78 to show that H is not closed under addition, it is sufficient to find two nilpotent matrices A and B such that (A+B)" 0 for all positive integers n.) [34] 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, 5 6 (Hint: to show that H is not closed under scalar multiplication, it is sufficient to find a real number r and a nilpotent matrix A such that (rA)" #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. choose
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![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 x 2 nilpotent matrices with real entries. Is H a subspace of the vector space V?
1. Does H contain the zero vector of V?
choose
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
[12] [56]
such as [[1,2], [3,4]], [[5,6], [7,8]] for the answer
[³ ¾), § 9]. (Hint:
3
78
to show that H is not closed under addition, it is sufficient to find two nilpotent
matrices A and B such that (A+B)" 0 for all positive integers n.)
[34]
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, 5 6
(Hint: to show that H is not closed under scalar multiplication, it is sufficient to
find a real number r and a nilpotent matrix A such that (rA)" #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.
choose](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fef19df16-a9cf-4d28-80b1-5d3bd6b485e3%2F660ff0b3-78d2-4714-a9a2-a09408e50d55%2Fji2f90w_processed.jpeg&w=3840&q=75)
Transcribed Image Text: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 x 2 nilpotent matrices with real entries. Is H a subspace of the vector space V?
1. Does H contain the zero vector of V?
choose
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
[12] [56]
such as [[1,2], [3,4]], [[5,6], [7,8]] for the answer
[³ ¾), § 9]. (Hint:
3
78
to show that H is not closed under addition, it is sufficient to find two nilpotent
matrices A and B such that (A+B)" 0 for all positive integers n.)
[34]
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, 5 6
(Hint: to show that H is not closed under scalar multiplication, it is sufficient to
find a real number r and a nilpotent matrix A such that (rA)" #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.
choose
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