Determine whether the given set S is a subspace of the vector space V.A. V is the vector space of all real-valued functions defined on the interval (−[infinity],[infinity]), and S is the subset of V consisting of those functions satisfying f(0)=0.B. V=Mn(R), and S is the subset of all nonsingular matrices.C. V=C3(I), and S is the subset of V consisting of those functions satisfying the differential equation y′′′+7y=x2.D. V=C5(I), and S is the subset of V consisting of those functions satisfying the differential equation y(5)=0.E. V=Pn, and S is the subset of Pn consisting of those polynomials satisfying p(0)=0.F. V=Mn(R), and S is the subset of all upper triangular matrices.G. V=P2, and S is the subset of P2 consisting of all polynomials of the form p(x)=x2+c.

kindly check the attached image below to Determine whether the given set S is a subspace of the vector space (which is contained within a different vector space. So all the subspace is a kind of vector space in their own way, although it is also defined relative to some of the other larger vector space. The linear subspace is more often than not simply called a subspace whenever the situation serves to differentiate it from other types of subspaces.) V.A