Determine whether the given set S is a subspace of the vecto

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=³, and S is the set of vectors (x₁,x₂,x₃) in V satisfying x₁6x₂+x₃=5.

C. V=ⁿ, and S is the set of solutions to the homogeneous linear system Ax=0 where A is a fixed m×n matrix.

D. V=C²(I), and S is the subset of V consisting of those functions satisfying the differential equation y4y+3y=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)=5.

F. V=P_n, and S is the subset of P_n consisting of those polynomials satisfying p(0)=0.

G. V=M_n(), and S is the subset of all symmetric matrices

Solution

A. no S is not a subspace since the 0-polynomial is not an element of S, since 0(1) = 0 = 0(0).

B. no S is not a subspace because it does not contain the 0-vector.

C. yes, S is a subspace

D. no, because (0,0) does not belongs to S.

E. no , because the 0-function 0:[a,b]-->R given by 0(x) = 0 for all x in [a,b] is not a member of S.

F. yes it is a subspace.

G. the 0-matrix is symmetric. suppose that A,B are symmetric matrices, so that

A^T = A and B^T = B. then (A+B)^T = A^T + B^T = A + B, so A+B is symmetric.

it is also clear that (cA)^T = c(A^T) = cA, so cA is symmetric as well. this is a subspace