Question

Determine whether the given set S is a subspace of the vector space V. Note: P_n(R) is the vector space of all real polynomials of degree at most n and M_n(R) is the vector space of all real n x n matrices A. 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). B. V = P_5(R), and S is the subset of V = P_5(R) consisting of those polynomials satisfying p(1) > p(0). C. V = C^3(I), and S is the subset of V consisting of those functions satisfying the differential equation y''' + 2y = x^2. D. V = M_n(R), and S is the subset of all skew-symmetric matrices. E. V = C^2(I), and S is the subset of V consisting of those functions satisfying the differential equation y'' - 4y' + 3y = 0. F. 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. G. V = R^5, and S is the set of vectors (x_1, x_2, x_3) in V satisfying x_1 - 4x_2 + x_3 = 3.

          Determine whether the given set S is a subspace of the vector space V. Note: P_n(R) is the vector space of all real polynomials of degree at most n and M_n(R) is the vector space of all real n x n matrices

A. 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).
B. V = P_5(R), and S is the subset of V = P_5(R) consisting of those polynomials satisfying p(1) > p(0).
C. V = C^3(I), and S is the subset of V consisting of those functions satisfying the differential equation y''' + 2y = x^2.
D. V = M_n(R), and S is the subset of all skew-symmetric matrices.
E. V = C^2(I), and S is the subset of V consisting of those functions satisfying the differential equation y'' - 4y' + 3y = 0.
F. 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.
G. V = R^5, and S is the set of vectors (x_1, x_2, x_3) in V satisfying x_1 - 4x_2 + x_3 = 3.

Determine whether the given set S is a subspace of the vector space V. Note: Pn(R) is the vector space of all real polynomials of degree at most n and Mn(R) is the vector space of all real n x n matrices

A. 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).
B. V = P5(R), and S is the subset of V = P5(R) consisting of those polynomials satisfying p(1) > p(0).
C. V = C^3(I), and S is the subset of V consisting of those functions satisfying the differential equation y”' + 2y = x^2.
D. V = Mn(R), and S is the subset of all skew-symmetric matrices.
E. V = C^2(I), and S is the subset of V consisting of those functions satisfying the differential equation y” - 4y' + 3y = 0.
F. 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.
G. V = R^5, and S is the set of vectors (x1, x2, x3) in V satisfying x1 - 4x2 + x3 = 3.

Added by Alex H.

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