The subset having diagonal elements nonzero. The subset of matrices whose (1, 2)

Question

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?

Explanation / Answer

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 P2.

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