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 Question 6
![A square matrix A is idempotent if A? = A.<br>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<br>matrices with real entries. Is H a subspace of the vector space<br>V?<br>1. Does H contain the zero vector of V?<br>H contains the zero vector of V<br>2. Is H closed under addition? If it is, enter CLOSED. If it is not, enter two matrices in H whose sum is<br>not in H, using a comma separated list and syntax such as [[1,2], [3,4]], [[5,6], [7,8]] for the<br>[1 21 [5 6]<br>answer<br>(Hint: to show that H is not closed under addition, it is sufficient to find two<br>3 4<br>7 8<br>idempotent matrices A and B such that (A + B)² + (A+ B).)<br>3. Is H closed under scalar multiplication? If it is, enter CLOSED. If it is not, enter a scalar in R and a<br>matrix in H whose product is not in H, using a comma separated list and syntax such as<br>[3 4]<br>2, [[3,4], [5,6]] for the answer 2, : (Hint: to show that H is not closed under scalar<br>5 6<br>multiplication, it is sufficient to find a real number r and an idempotent matrix A such that<br>(rA)² + (rA).)<br>4. Is H a subspace of the vector space V? You should be able to justify your answer by writing a<br>complete, coherent, and detailed proof based on your answers to parts 1-3.<br>H is not a subspace of V<br>](https://s3.us-east-1.amazonaws.com/storage.unifolks.com/qimg-008/008_e9t4xp0-qv4bxy5b.png)
Extracted text: 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 separated list and syntax such as [[1,2], [3,4]], [[5,6], [7,8]] for the [1 21 [5 6] answer (Hint: to show that H is not closed under addition, it is sufficient to find two 3 4 7 8 idempotent matrices A and B such that (A + B)² + (A+ B).) 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 4] 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 an idempotent matrix A such that (rA)² + (rA).) 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