Determine whether the given set S is a subspace of the vector space V Note: ?n(?

Question

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

A. V=?3(?), and S is the subset of V=?3(?) consisting of all polynomials of the form p(x)=ax3+bx.
B. V=?5(?), and S is the subset of V=?5(?) consisting of those polynomials satisfying p(1)>p(0).
C. V=Mn(?), and S is the subset of all invertible matrices.
D. V=C1(?), 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=?n, and S is the set of solutions to the homogeneous linear system Ax=0where A is a fixed m

Explanation / Answer

A. Test 1: Does S contain 0?
Answer: Yes. The polynomial p(x) = 0 is of the form ax^3 + bx, with a = b = 0. So it's in S.

Test 2: If p(x) and q(x) are in S, is r*p(x) + s*q(x) in the space?
Answer: Let p(x) = a1*x^3 + b1*x and q(x) = a2*x^3 + b2*x
Then r*p(x) + s*q(x) = (r*a1 + s*a2) x^3 + (r*b1 + s*b2) x which is of the form ax^3 + bx. So it's a member of S.

Therefore S is a vector space, i.e it's a subspace of V.

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