The subset having diagonal elements nonzero The subset of ma

The subset having diagonal elements nonzero. The subset of matrices whose (1, 2) element is 0. (For example [2 0 3 4] would be such a matrix.) The subset of matrices of the form [a b -b c]. The subset of 2 times 2 matrices whose determinants are zero. Let S be the set of all functions of the form f(x) = 3 x^2 + ax - b, where a and b are real numbers. Is S a subspace of P_2? Is the condition that a subset of a vector space contain the zero vector a necessary and sufficient condition for the subset to be a subspace?

Solution

Let f(x) = 3x² +ax –b and g(x)= 3x² +cx –d be two arbitrary elemennts of S where a, b, c and and d are real numbers. Then f(x) +g(x) = 3x² +ax –b +3x² +cx –d = 6x² + (a+c) x –(b+d). Since the coefficient of x² in f(x) +g(x) is not 3, hence f(x)+ g(x) does not belong to S. Thus, S is not closed under vector addition and ,therefore, S is not a vector space. Hence S is not a subspace of P_2.