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 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

the transverse axis of a hyperbola is the line that contains the two vertices and the two focuses.

in this example (hyperbola of equation x^2/2 - y^2/4 = 1), the transverse axis is the  x-axis  . 

step-by-step explanation:

7! = 7×6×5×4×3×2

3! = 3× 2

as 7! is multiple of 3! because 7! contains 3! that is 3× 2 as factor

so 7! /(3! ) = 7× 6× 5× 4

= 42 × 20

= 840

its b) 840✌✌✌

dhruv⭐