Question

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

A. V={\mathbb R}^2 , and S consists of all vectors (x_1, x_2) satisfyingx_1^2 - x_2^2 = 0.

B. V=M_n({\mathbb R}) , and S is the subset of all symmetric matrices
C. V=P_n , and S is the subset of P_n consisting of those polynomials satisfying p(0) = 0.
D. V=C^2(I) , and S is the subset of V consisting of those functions satisfying the differential equation y''-4y'+3y=0.
E. V is the vector space of all real-valued functions defined on the interval(-\infty, \infty), and S is the subset of V consisting of those functions satisfyingf(0)=0.
F. 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)=4.
G. V=C^3(I) , and S is the subset of V consisting of those functions satisfying the differential equation y'''+3 y=x^2.

Answer 1

Answer and Step-by-step explanation:

A) False, take (1,-1) and (1,1) both in V, but the sum (2,0) is not

B) True, if A,B are symmetric then A+kB is symmetric as well for k scalar, and zero matrix is symmetric.

C) True. The zero polynom is in V and P+kQ verifies (P+kQ)(0)=P(0)+kQ(0)=0

D) True. zero is solution and f+kg is since all coefficients are linear.

E) True, same than C)

F) False , the zero function doesn't verify that ( and the closure for sum will fail anyway 4+4=8)

G) False, the zero function fail to satisfy this ODE