If H is idempotent, is and V is a vector space of all 2x2 matricies with real en
ID: 3004773 • Letter: I
Question
If H is idempotent, is and V is a vector space of all 2x2 matricies with real entires, is H a subspace of V?
I'm not certain where I went wrong with my answers below.
(1 point) A square matrix A is idempotent if AA. Let V be the vector space of all 2 × 2 matrices with real entries. Let H be the set of all 2 × 2 idempotent matrices with real entries. ls 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 S 6.(Hint: to show that H is not 1 2] [56 3 4]'17 8 separated list and syntax such as 11.21.13,411.1I5.61.17.8]) for the answer 3 4 closed under addition, it is sufficient to find two idempotent matrices A and B such that (A + B)2 (A + B) CLOSED 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, not closed under scalar multiplication, it is sufficient to find a real number and an idempotent matrix A such that (rA)2 (rA)) 3. (Hint: to show that H is 2, I1,01,10,1 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. 4. H is not a subspace of VExplanation / Answer
1) Yes, H contains the EigenValues of any 2*2 vector V
2) You can take A and B, both to be I matrix and see what happens with addition
A + B = I + I = 2I. For additon to be closed over H, we need (A + B)^ 2 = A + B
(A + B)2 = (2I)^2 = 4I and since that's not same as A + B, H is not closed under addition.
In fact, (A+B)^2 = A2 + 2AB + B2 = A + B + 2AB. If you can find idempotent matrices A and B, such that AB != 0,
then H won' t be closed!
3) (rA)2 = r2A2 = r2 A
r2A = rA = > r = 1. Hence any real number r not 1 violates the conditon for closed multiplication on H
4) Clearly, H is a subset of V as V is set of all 2*2 real matrices. H is a subspace of V iif (if and only if) it is closed on addition and scalar multiplication operations on V. We know that V as, it's a commutative ring but H is not even a vector space as it doesn't satifsty aforementioned closed criteria.
Thus, H is not a subspace of V.