Need help with vector space multiple choice

1. Jan 21, 2010

neilpeart0408

Determine whether the given set S is a subspace of the vector space V.

A. V=P5, and S is the subset of P5 consisting of those polynomials satisfying p(1)>p(0).
B. 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).
C. V=ℝn, and S is the set of solutions to the homogeneous linear system Ax=0 where A is a fixed m×n matrix.
D. V=Mn(ℝ), and S is the subset of all n×n matrices with det(A)=0.
E. V=P4, and S is the subset of P4 consisting of all polynomials of the form p(x)=ax3+bx.
F. V=C2(I), and S is the subset of V consisting of those functions satisfying the differential equation yʺ=0.
G. V=ℝ3, and S is the set of vectors (x1,x2,x3) in V satisfying x1-9x2+x3=8.

2. Jan 21, 2010

VeeEight

S is a subspace of V if, (1) S is closed under addition (given two elements a and b in S, a+b is in S) and (2)S is closed under scalar multiplication (given x in S, and a scalar c, cx is in S).

For each question you should see if the set satisfies these two criteria. If it doesn't then find a counterexample.

For example problem D, does detA = 0 = detB mean that det(A+B) = 0?