Math  /  Algebra

QuestionA square matrix AA is idempotent if A2=AA^{2}=A. Let VV be the vector space of all 2×22 \times 2 matrices with real entries. Let HH be the set of all 2×22 \times 2 idempotent matrices with real entries. Is HH a subspace of the vector space VV ?
1. Does HH contain the zero vector of VV ? choose
2. Is HH closed under addition? If it is, enter CLOSED. If it is not, enter two matrices in HH whose sum is not in HH, using a comma separated list and syntax such as [[1,2],[3,4]],[[5,6],[7,8]][[1,2],[3,4]],[[5,6],[7,8]] for the answer [1234],[5678]\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right],\left[\begin{array}{ll}5 & 6 \\ 7 & 8\end{array}\right]. (Hint: to show that HH is not closed under addition, it is sufficient to find two idempotent matrices AA and BB such that (A+B)2(A+B)(A+B)^{2} \neq(A+B).) \square
3. Is HH closed under scalar multiplication? If it is, enter CLOSED. If it is not, enter a scalar in R\mathbb{R} and a matrix in HH whose product is not in HH, using a comma separated list and syntax such as 2,[[3,4],[5,6]]2,[[3,4],[5,6]] for the answer 2,[3456]2,\left[\begin{array}{ll}3 & 4 \\ 5 & 6\end{array}\right]. (Hint: to show that HH is not closed under scalar multiplication, it is sufficient to find a real number rr and an idempotent matrix AA such that (rA)2(rA)(r A)^{2} \neq(r A).) \square
4. Is HH a subspace of the vector space VV ? 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

Studdy Solution
Determine if H H is a subspace of V V .
Based on the previous steps: - H H contains the zero vector of V V . - H H is not closed under addition. - H H is not closed under scalar multiplication.
Since H H is not closed under addition and scalar multiplication, it is not a subspace of V V .
The set H H is not a subspace of the vector space V V .

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