Determine whether the given set S is a subspacc of the vector space V. V = Mn(R)

Question

Determine whether the given set S is a subspacc of the vector space V. V = Mn(R), and S is the subset of all skew-symmetric matrices. V = C2(I), and S is the subset of V consisting of those functions satisfying the differential equation y" - Ay' + 3y = 0. V = P3, and S is the subset of P3 consisting of all polynomials of the form p(x) = ax3 + bx. V = P5, and S is the subset of P5 consisting of those polynomials satisfying p(1) > p(0). V = R2, and S is the set of all vectors (x1, x2) in V satisfying 5x1 + 6x2 = 0. V = C3(I), and S is the subset of V consisting of those functions satisfying the differential equation if''' + 4 y = x2. 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) = 5.

Explanation / Answer

(a) Subspace.
(b) Subspace.
(c) Not a subspace. Does not contain zero (unless a,b are allowed to be 0, otherwise it is a subpace).
(d) Not a subspace. Does not contain zero. 0(1) = 0(0).
(e) Subspace. Zero is obviously in there, as are additive inverses. Closure is trivial.
(f) Not a subspace. Does not contain zero.
(g) Not a subspace. Obviously not closed under addition or scalar multiplication.

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