Determine whether the given set S is a subspace of the vector space V. A. V=P3, and S is the subset of P3 consisting of all polynomials of the form p(x)=ax^3+bx. B. V=P5, and S is the subset of P5 consisting of those polynomials satisfying p(1)>p(0). C. V=ℝ^n×n, and S is the subset of all nonsingular matrices. D. V=C1(ℝ), and S is the subset of V consisting of those functions satisfying f′(0)≥0. E. 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). F. V=ℝ^n, and S is the set of solutions to the homogeneous linear system Ax=0 where A is a fixed m×n matrix. G. V is the space of twice differentiable functions ℝ→ℝ, and S is the subset of V consisting of those functions satisfying the differential equation y″−4y′+3y=0.
Determine whether the given set S is a subspace of the
A. V=P3, and S is the subset of P3 consisting of all polynomials of the form p(x)=ax^3+bx.
B. V=P5, and S is the subset of P5 consisting of those polynomials satisfying p(1)>p(0).
C. V=ℝ^n×n, and S is the subset of all nonsingular matrices.
D. V=C1(ℝ), and S is the subset of V consisting of those functions satisfying f′(0)≥0.
E. 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).
F. V=ℝ^n, and S is the set of solutions to the homogeneous linear system Ax=0 where A is a fixed m×n matrix.
G. V is the space of twice differentiable functions ℝ→ℝ, and S is the subset of V consisting of those functions satisfying the differential equation y″−4y′+3y=0.
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