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.

Question

Douglas Parrish

Mathematics

2 November, 21:02

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.

+2

Answers (1)

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Marisol Miranda · 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