Determine whether the given set S is a subspace of the vector space V. V = P_3,

Question

Determine whether the given set S is a subspace of the vector space V. V = P_3, and S is the subset of P_3 consisting of all polynomials of the form p(x) = x_2 + c. V = P_n, and S is the subset of P_n consisting of those polynomials satisfying p(0) = 0. V = M_n (R), and S is the subset of all symmetric matrices V = M_n (R), and S is the subset of all nonsingular matrices. V = R^3, and S is the set of vectors (x_1, x_2, x_3) in V satisfying x_1 - 7x_2 + x_3 = 6. V is the vector space of all real-valued functions defined on the interval (- infinity, infinity), and S is the subset of V consisting of those functions satisfying f(0) = 0. V = R_n, and S is the set of solutions to the homogeneous linear system Ax = 0 where A is a fixed m Times n matrix.

Explanation / Answer

g)yes. classic. if v,w belong to S. then Av=0 and Aw=0.
adding these 2 you get Av + Aw = 0
therefore by laws of matrix multiplcation: A(v+w)=0 therefore v+w belong to S.
Also: Av=0 then aAv=0 where a is a scalar. therefore A(av)=0 therefore av also belongs to S

c) & d) & e) yes

f) & b) yes. because if v,w belong to S so does v+w and a*v (where a belongs to F) belong to S. If you can't "see this", try to add 2 polynomials satisfying p(0) = 0 , the result would always be a 3rd polynomial that also satisfyies p(0)=0. same goes with scalar multiplication (ie a*v).

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