Prove the S is a subspace of M22 S a 0 0 b M22 ab RSolution

Prove the S is a subspace of M(2,2) S = {[a 0 0 b] M(2,2)| a,b R}

Solution

0 H? Is the zero vector 0 in V also a member of H? First think what exactly is meant by the symbol 0 in the vector space S . In M2×2, the symbol 0 stands for the matrix 0 0 0 0 . In C[a, b], the vector space of all functions f : [a, b] R that are continuous on the interval [a, b], the symbol 0 stands for the constant function whose output is the number 0 for every input: a t b = 0(t) = 0. Your proof should begin with the sentence In the vector space V , the zero vector 0 is hexplicit description of 0 herei. Think. Does the 0 in V satisfy the H-rule? Write a sentence explaining why it does or does not. Be specific. 0 is one particular member of V . Either continue your proof with the sentence Since hexplain that 0 satisfies H-rulei, 0 is an element of H. or finish it with the sentence Since hexplain that 0 violates H-rulei, 0 is not in H, and H is not a subspace of V .