Determine whether the given set S is a subspace of the vector space V. A. V=R^n,

Question

Determine whether the given set S is a subspace of the vector space V. A. V=R^n, and S is the set of solutions to the homogeneous linear system Ax=0 where A is a fixed m x n matrix. B. V=R^4, and S is the set of vectors of the form (0, x2, 7, x4) C. V=C^1(R), and S is the subset of V consisting of those functions satisfying f'(0)?0 D. V=C^2(I), and S is the subset of V consisting of those functions satisfying the differential equation y''-4y'+3y=0 E. V=P5, and S is the subset of P5 consisting of those polynomials satisfying p(1)>p(0) F. V is the vector space of all real-valued functions defined on the interval[a,b], and S is the subset of V consisting of those functions satisfying f(a) = f(b) G. V=P4, and S is the subset of P4 consisting of all polynomials of the form p(x)=ax^3 + bx I thought the answers would be A,C,D but they weren't correct. will somebody please help me out understanding this question?

Explanation / Answer

A, B. No, doesn't contain zero. G. No, not closed under multiplication by -1. C, D, E. Yes, these sets are kernels of a linear operator. F. Yes, it's the span of {x, x^3}. s

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